0,2

A007097(n) = Min {k : A109301(k) = n} = the first k whose rote height is n, the level set leader or minimum inverse function corresponding to A109301. - Jon Awbrey (jawbrey(AT)att.net), Jun 26 2005

Lubomir Alexandrov (alexandr(AT)theor.jinr.ru) informs me that he studied this sequence in his 1965 notebook. - N. J. A. Sloane (njas(AT)research.att.com), May 23 2008.

Lubomir Alexandrov, unpublished notes, circa 1960.

Lubomir Alexandrov, "On the nonasymptotic prime number distribution", LANL.math.NT/9811096, Los Alamos, 1998

Lubomir Alexandrov, "Prime Number Sequences And Matrices Generated By Counting Arithmetic Functions", Communications of the Joint Institute of Nuclear Research, E5-2002-55, Dubna, 2002.

Lubomir Alexandrov,"The Eratosthenes Progression p(k+1)=p(p(k)), k=0,1,2,..., p(0)=1,4,6,... Determines an Inner Prime Number Distribution Law", Second Int. Conf. "Modern Trends in Computational Physics", Jul 24-29, 2000, Dubna, Russia, Book of Abstracts, p. 19.

L. Longeri, Towards understanding nature and the aesthetics of prime numbers, https://www.longeri.org/prime/nature.html

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Deleglise, Computation of large values of pi(x)

N. Fernandez, An order of primeness, F(p)

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

NestList[Prime@# &, 1, 16] (from Robert G. Wilson v (rgwv(at)rgwv.com), May 30 2006)

Cf. A000720, A049076-A049081.

Cf. A109301.

Sequence in context: A090709 A112279 A130166 this_sequence A132745 A124538 A124627

Adjacent sequences: A007094 A007095 A007096 this_sequence A007098 A007099 A007100

nonn,hard,nice

N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com).

15th term corrected and 2 more terms added by loria.fr!Paul.Zimmermann (Paul Zimmermann).

a(18) and a(19) from David Baugh (dbaugh(AT)rice.edu), Oct 25 2007

0,3

Also called the Bell triangle or the Pierce triangle.

Let P be the lower-triangular Pascal-matrix, Then this is exp(P) / exp(1). - Gottfried Helms (helms(AT)uni-kassel.de), Mar 30 2007.

a(n,k) is the number of equivalence relations on {0, ..., n} such that k is not equivalent to n, k+1 is not equivalent to n, ..., n-1 is not equivalent to n. - D. E. Knuth, Sep 21, 2002. [Comment revised by Thijs van Ommen (thijsvanommen(AT)gmail.com), Jul 13 2008]

A. C. Aitken, A problem on combinations, Edinburgh Math. Notes 28 (1933), 18-33.

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 205.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 212.

D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.5.

Charles Sanders Peirce, On the Algebra of Logic, American Journal of Mathematics, Vol. 3, pages 15-57, 1880. Reprinted in Collected Papers (1935-1958) and in Writings of Charles S. Peirce: A Chronological Edition (Indiana University Press, Bloomington, IN, 1986).

T. D. Noe, Rows n=0..50 of triangle, flattened

D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Charles Sanders Peirce, Works

Charles Sanders Peirce, Collected Papers

Eric Weisstein's World of Mathematics, Bell Triangle

Nick Hobson, Python program for this sequence

Double-exponential generating function: sum_{n, k} a(n-k, k) x^n y^k / n! k! = exp(e^{x+y}-1+x). - D. E. Knuth, Sep 21, 2002. [U coordinates, reversed]

a(n,k) = Sum_{i=0..k} binomial(k,i)*Bell(n-k+i). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 15 2006

Triangle begins:

1;

1,2;

2,3,5;

5,7,10,15;

15,20,27,37,52;

...

A011971 := proc(n, k) option remember; if n=0 and k=0 then 1 elif k=0 then A011971(n-1, n-1) else A011971(n, k-1)+A011971(n-1, k-1); fi: end;

for n from 0 to 12 do lprint([ seq(A011971(n, k), k=0..n) ]); od:

a[0, 0] = 1; a[n_, 0] := a[n - 1, n - 1]; a[n_, k_] := a[n, k - 1] + a[n - 1, k - 1]; Flatten[ Table[ a[n, k], {n, 0, 9}, {k, 0, n}]] (from Robert G. Wilson v Mar 27 2004)

Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, etc., A011968, A011969. Cf. A046934, A011972 (duplicates removed).

Main diagonal is in A094577. Mirror image is in A123346.

See also A095149, A106436, A108041, A108042, A108043.

Sequence in context: A033189 A008507 A028364 this_sequence A060048 A110699 A035537

Adjacent sequences: A011968 A011969 A011970 this_sequence A011972 A011973 A011974

tabl,nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), J. H. Conway and R. K. Guy

Peirce reference from Jon Awbrey (jawbrey(AT)att.net), Mar 11, 2002

0,3

J. Awbrey, personal journal, circa 1978. Letter to N. J. A. Sloane, 1980-Aug-04.

V. Jovovic, Table of n, a(n) for n=0..100

J. Awbrey, Illustration of initial terms

V. Jovovic, First 100 terms

G.f. A(x) = 1 + x + 2*x^2 + 6*x^3 + ... satisfies A(x) = Product_{j = 0 to infinity} (1 + x^(j+1)*A(x))^a_j.

These structures come from recursive primes' factorizations of natural numbers, where the recursion proceeds on both the exponents (^k) and the indices (_k) of the primes invoked in the factorization:

2 = (prime_1)^1 = (p_1)^1, briefly, p, weight of 1 node => a(1) = 1.

3 = (prime_2)^1 = (p_2)^1, briefly, p_p, weight of 2 nodes and

4 = (prime_1)^2 = (p_1)^2, briefly, p^p, weight of 2 nodes => a(2) = 2.

a(0) := 1: for k from 1 to 30 do A := add(a(i)*x^i, i=0..k): B := mul((1+x^(j+1)*A)^a(j), j=0..k-1): a(k) := coeff(series(B, x, k+1), x, k): printf(`%d, `, a(k)); od:

Cf. A062504, A062860.

Sequence in context: A150139 A052884 A150140 this_sequence A104632 A150141 A150142

Adjacent sequences: A061393 A061394 A061395 this_sequence A061397 A061398 A061399

nice,nonn,easy

Jon Awbrey (jawbrey(AT)oakland.edu), Jun 09 2001

Corrected and extended with Maple program by Vladeta Jovovic and David W. Wilson (davidwwilson(AT)comcast.net), Jun 20 2001

0,2

k | natural numbers n such that |riff(n)| = k

0 | 1;

1 | 2;

2 | 3, 4;

3 | 5, 6, 7, 8, 9, 16;

4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536;

The natural number values for the riffs with at most 3 pts are as follows (x = root):

.................o.......o..o.......o

.................|.......^..|.......^

.................v.......|..v.......|

...........o..o..o....o..o..o..o.o..o

...........|..^..|....|..|..^..|.^..^

...........v..|..v....v..v..|..v/...|

Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;

Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;

Cf. A061396, A062860.

Sequence in context: A048406 A081433 A032581 this_sequence A102489 A004838 A032998

Adjacent sequences: A062501 A062502 A062503 this_sequence A062505 A062506 A062507

nonn

Jon Awbrey (jawbrey(AT)oakland.edu), Jun 23 2001

1,3

A061396(n) gives number of times n appears in this sequence.

J. Awbrey, Illustrations of riffs for small integers

a(Prod(p_i^e_i)) = Sum(a(i)+a(e_i)+1), product over nonzero e_i in prime factorization.

Sequence in context: A072643 A130260 A111393 this_sequence A097688 A025785 A087182

Adjacent sequences: A062534 A062535 A062536 this_sequence A062538 A062539 A062540

nonn,easy,nice

David W. Wilson (davidwwilson(AT)comcast.net), Jun 25, 2001

0,2

Greatest j with n nodes in its riff is given by A014221(n+1).

Cf. A014221, A061396, A062504, A062537, A111800.

Sequence in context: A133335 A062925 A118728 this_sequence A024369 A047011 A018163

Adjacent sequences: A062857 A062858 A062859 this_sequence A062861 A062862 A062863

nonn

Jon Awbrey (jawbrey(AT)oakland.edu), Jun 25 2001

More terms from David W. Wilson (davidwwilson(AT)comcast.net), Jun 26, 2001

1,5

The right diagonal labeled by the prime power of the form j:k = (prime(j))^k contains the j^th power primes in the factorization raised to the k^th power. For example, the right diagonal labeled by the number 2 = 1:1 = (prime(1))^1 contains the power-free parts of each positive integer, specifically A055231 and the right diagonal labeled by the number 4 = 1:2 = (prime(1))^2 contains the squares of the square-free parts of positive integers.

In general, then the right diagonal labeled by m = (j_i : k_i)_i = Product_i prime(j_i)^(k_i) contains the product over i of the (j_i)th power primes in the factorization raised to the (k_i)th powers.

For example, the operator 5 = 3:1 extracts the 3rd power primes in the factorization of each n and raises them to the first power, thus sending 8 = 1:3 to 2 = 1:1, 27 = 2:3 to 3 = 2:1 and so on.

` ` ` ` ` ` ` ` ` ` `n o m

` ` ` ` ` ` ` ` ` ` ` \ /

` ` ` ` ` ` ` ` ` ` `1 . 1

` ` ` ` ` ` ` ` ` ` \ / \ /

` ` ` ` ` ` ` ` ` `2 . 1 . 2

` ` ` ` ` ` ` ` ` \ / \ / \ /

` ` ` ` ` ` ` ` `3 . 1 . 1 . 3

` ` ` ` ` ` ` ` \ / \ / \ / \ /

` ` ` ` ` ` ` `4 . 1 . 2 . 1 . 4

` ` ` ` ` ` ` \ / \ / \ / \ / \ /

` ` ` ` ` ` `5 . 1 . 3 . 1 . 1 . 5

` ` ` ` ` ` \ / \ / \ / \ / \ / \ /

` ` ` ` ` `6 . 1 . 1 . 1 . 4 . 1 . 6

` ` ` ` ` \ / \ / \ / \ / \ / \ / \ /

` ` ` ` `7 . 1 . 5 . 2 . 9 . 1 . 1 . 7

` ` ` ` \ / \ / \ / \ / \ / \ / \ / \ /

` ` ` `8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8

` ` ` \ / \ / \ / \ / \ / \ / \ / \ / \ /

` ` `9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9

` ` \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /

` 10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10

Primal codes of finite partial functions on positive integers:

1 = { }

2 = 1:1

3 = 2:1

4 = 1:2

5 = 3:1

6 = 1:1 2:1

7 = 4:1

8 = 1:3

9 = 2:2

10 = 1:1 3:1

11 = 5:1

12 = 1:2 2:1

13 = 6:1

14 = 1:1 4:1

15 = 2:1 3:1

16 = 1:4

17 = 7:1

18 = 1:1 2:2

19 = 8:1

20 = 1:2 3:1

Cf. A061396, A062504, A062537, A062860, A106178.

Sequence in context: A147867 A114118 A146014 this_sequence A135010 A138138 A080209

Adjacent sequences: A106174 A106175 A106176 this_sequence A106178 A106179 A106180

nonn,tabl

Jon Awbrey (jawbrey(AT)att.net), May 23 2005

1,3

Table of Rotes and Primal Functions for Positive Integers from 1 to 40

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` o-o ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` `

` ` ` ` ` ` ` o-o ` ` ` ` ` o-o ` ` ` ` ` o-o ` ` ` ` ` o-o ` ` ` ` ` `

` ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` `

O ` ` ` ` ` ` O ` ` ` ` ` ` O ` ` ` ` ` ` O ` ` ` ` ` ` O ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

{ } ` ` ` ` ` 1:1 ` ` ` ` ` 2:1 ` ` ` ` ` 1:2 ` ` ` ` ` 3:1 ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

1 ` ` ` ` ` ` 2 ` ` ` ` ` ` 3 ` ` ` ` ` ` 4 ` ` ` ` ` ` 5 ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` o-o ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` `

` ` ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `

` ` o-o ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` o-o o-o ` ` ` ` ` o-o ` ` ` `

` ` | ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` | ` | ` ` ` ` ` ` | ` ` ` ` `

o-o o-o ` ` ` o-o ` ` ` ` ` o-o ` ` ` ` ` o---o ` ` ` ` o-o o-o ` ` ` `

| ` | ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` | ` | ` ` ` ` `

O===O ` ` ` ` O ` ` ` ` ` ` O ` ` ` ` ` ` O ` ` ` ` ` ` O===O ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

1:1 2:1 ` ` ` 4:1 ` ` ` ` ` 1:3 ` ` ` ` ` 2:2 ` ` ` ` ` 1:1 3:1 ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

6 ` ` ` ` ` ` 7 ` ` ` ` ` ` 8 ` ` ` ` ` ` 9 ` ` ` ` ` ` 10` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

o-o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

o-o ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` o-o ` ` ` `

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` `

o-o ` ` ` ` ` ` o-o o-o ` ` o-o o-o ` ` ` ` ` o-o ` ` ` o-o o-o ` ` ` `

| ` ` ` ` ` ` ` | ` | ` ` ` | ` | ` ` ` ` ` ` | ` ` ` ` | ` | ` ` ` ` `

o-o ` ` ` ` ` o-o ` o-o ` ` o===o-o ` ` ` o-o o-o ` ` ` o-o o-o ` ` ` `

| ` ` ` ` ` ` | ` ` | ` ` ` | ` ` ` ` ` ` | ` | ` ` ` ` | ` | ` ` ` ` `

O ` ` ` ` ` ` O=====O ` ` ` O ` ` ` ` ` ` O===O ` ` ` ` O===O ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

5:1 ` ` ` ` ` 1:2 2:1 ` ` ` 6:1 ` ` ` ` ` 1:1 4:1 ` ` ` 2:1 3:1 ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

11` ` ` ` ` ` 12` ` ` ` ` ` 13` ` ` ` ` ` 14` ` ` ` ` ` 15` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `

` ` o-o ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` o-o ` ` `

` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` `

` o-o ` ` ` ` o-o ` ` ` ` ` ` ` o-o o-o ` o-o ` ` ` ` ` ` o-o o-o ` ` `

` | ` ` ` ` ` | ` ` ` ` ` ` ` ` | ` | ` ` | ` ` ` ` ` ` ` | ` | ` ` ` `

o-o ` ` ` ` ` o-o ` ` ` ` ` o-o o---o ` ` o-o ` ` ` ` ` o-o ` o-o ` ` `

| ` ` ` ` ` ` | ` ` ` ` ` ` | ` | ` ` ` ` | ` ` ` ` ` ` | ` ` | ` ` ` `

O ` ` ` ` ` ` O ` ` ` ` ` ` O===O ` ` ` ` O ` ` ` ` ` ` O=====O ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

1:4 ` ` ` ` ` 7:1 ` ` ` ` ` 1:1 2:2 ` ` ` 8:1 ` ` ` ` ` 1:2 3:1 ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

16` ` ` ` ` ` 17` ` ` ` ` ` 18` ` ` ` ` ` 19` ` ` ` ` ` 20` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` o-o ` ` ` ` o-o ` ` ` o-o o-o ` ` ` ` o-o ` ` ` ` o-o ` ` ` ` ` `

` ` ` | ` ` ` ` ` | ` ` ` ` | ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` `

o-o o-o ` ` ` ` ` o-o ` ` ` o---o ` ` ` ` ` o-o o-o ` ` o-o o-o ` ` ` `

| ` | ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` | ` | ` ` ` | ` | ` ` ` ` `

o-o o-o ` ` ` o-o o-o ` ` ` o-o ` ` ` ` ` o-o ` o-o ` ` o---o ` ` ` ` `

| ` | ` ` ` ` | ` | ` ` ` ` | ` ` ` ` ` ` | ` ` | ` ` ` | ` ` ` ` ` ` `

O===O ` ` ` ` O===O ` ` ` ` O ` ` ` ` ` ` O=====O ` ` ` O ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

2:1 4:1 ` ` ` 1:1 5:1 ` ` ` 9:1 ` ` ` ` ` 1:3 2:1 ` ` ` 3:2 ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

21` ` ` ` ` ` 22` ` ` ` ` ` 23` ` ` ` ` ` 24` ` ` ` ` ` 25` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` o-o ` ` ` o-o ` ` ` ` ` ` ` o-o ` ` ` o-o ` ` ` ` ` ` ` o-o ` `

` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` | ` ` `

` ` o-o o-o ` o-o o-o ` ` ` ` o-o o-o ` ` o-o o-o ` ` ` ` ` o-o o-o ` `

` ` | ` | ` ` | ` | ` ` ` ` ` | ` | ` ` ` | ` | ` ` ` ` ` ` | ` | ` ` `

o-o o===o-o ` o---o ` ` ` ` o-o ` o-o ` ` o===o-o ` ` ` o-o o-o o-o ` `

| ` | ` ` ` ` | ` ` ` ` ` ` | ` ` | ` ` ` | ` ` ` ` ` ` | ` | ` | ` ` `

O===O ` ` ` ` O ` ` ` ` ` ` O=====O ` ` ` O ` ` ` ` ` ` O===O===O ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

1:1 6:1 ` ` ` 2:3 ` ` ` ` ` 1:2 4:1 ` ` ` 10:1` ` ` ` ` 1:1 2:1 3:1 ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

26` ` ` ` ` ` 27` ` ` ` ` ` 28` ` ` ` ` ` 29` ` ` ` ` ` 30` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

o-o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

o-o ` ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` `

| ` ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `

o-o ` ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` ` o-o ` ` ` o-o ` o-o ` ` `

| ` ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` | ` ` | ` ` ` `

o-o ` ` ` ` ` ` o-o ` ` ` ` o-o o-o ` ` ` ` ` o-o ` ` ` o-o o-o ` ` ` `

| ` ` ` ` ` ` ` | ` ` ` ` ` | ` | ` ` ` ` ` ` | ` ` ` ` | ` | ` ` ` ` `

o-o ` ` ` ` ` o-o ` ` ` ` ` o-o o-o ` ` ` o-o o-o ` ` ` o-o o-o ` ` ` `

| ` ` ` ` ` ` | ` ` ` ` ` ` | ` | ` ` ` ` | ` | ` ` ` ` | ` | ` ` ` ` `

O ` ` ` ` ` ` O ` ` ` ` ` ` O===O ` ` ` ` O===O ` ` ` ` O===O ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

11:1` ` ` ` ` 1:5 ` ` ` ` ` 2:1 5:1 ` ` ` 1:1 7:1 ` ` ` 3:1 4:1 ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

31` ` ` ` ` ` 32` ` ` ` ` ` 33` ` ` ` ` ` 34` ` ` ` ` ` 35` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` o-o o-o ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` o-o o-o ` ` `

` ` ` ` ` ` ` ` | ` | ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` | ` | ` ` ` `

` o-o o-o o-o o-o ` o-o ` ` ` ` o-o ` ` ` o-o o-o o-o ` ` o-o o-o ` ` `

` | ` | ` | ` | ` ` | ` ` ` ` ` | ` ` ` ` | ` | ` | ` ` ` | ` | ` ` ` `

o-o ` o---o ` o=====o-o ` ` o-o o-o ` ` ` o-o o===o-o ` o-o ` o-o ` ` `

| ` ` | ` ` ` | ` ` ` ` ` ` | ` | ` ` ` ` | ` | ` ` ` ` | ` ` | ` ` ` `

O=====O ` ` ` O ` ` ` ` ` ` O===O ` ` ` ` O===O ` ` ` ` O=====O ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

1:2 2:2 ` ` ` 12:1` ` ` ` ` 1:1 8:1 ` ` ` 2:1 6:1 ` ` ` 1:3 3:1 ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

36` ` ` ` ` ` 37` ` ` ` ` ` 38` ` ` ` ` ` 39` ` ` ` ` ` 40` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

In these Figures, "extended lines of identity" like o===o

indicate identified nodes and capital O is the root node.

The rote height in gammas is found by finding the number

of graphs of the following shape between the root and one

of the highest nodes of the tree:

o--o

|

o

A sequence like this, that can be regarded as a nonnegative integer

measure on positive integers, may have as many as 3 other sequences

associated with it. Given that the fiber of a function f at n is all

the domain elements that map to n, we always have the fiber minimum

or minimum inverse function and may also have the fiber cardinality

and the fiber maximum or maximum inverse function. For A109301, the

minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the

first positive integer whose rote height is n, the fiber cardinality

is A109300, giving the number of positive integers of rote height n,

while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving

the last positive integer whose rote height is n, has the following

initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36,

while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 =

(2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly

7.840858554516122655953405327738 x 10^371.

Writing (prime(i))^j as i:j, the primes factorization of a positive integer n can be written as n = prod_(k = 1 to m) i(k):j(k). This sets up the formula: rhig(n) = 1 + max_(k = 1 to m) {rhig(i(k)), rhig(j(k))}, where rhig(1) = 0.

Writing (prime(i))^j as i:j, we have:

802701 = 2:2 8638:1

8638 = 1:1 4:1 113:1

113 = 30:1

30 = 1:1 2:1 3:1

4 = 1:2

3 = 2:1

2 = 1:1

1 = { }

So rote(802701) is the graph:

` ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` o-o

` ` ` ` ` ` ` ` ` ` ` ` ` | `

` ` ` ` ` ` ` ` ` ` ` o-o o-o

` ` ` ` ` ` ` ` ` ` ` | ` | `

` ` ` ` ` ` ` o-o o-o o-o o-o

` ` ` ` ` ` ` | ` | ` | ` | `

` ` ` ` ` ` o-o ` o===o===o-o

` ` ` ` ` ` | ` ` | ` ` ` ` `

o-o o-o o-o o-o ` o---------o

| ` | ` | ` | ` ` | ` ` ` ` `

o---o ` o===o=====o---------o

| ` ` ` | ` ` ` ` ` ` ` ` ` `

O=======O ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Therefore rhig(802701) = 6.

Cf. A007097, A050924, A061396, A062504, A062537, A062860.

Cf. A106177, A108352, A108371, A109300, A111791 to A111800.

Sequence in context: A072832 A080328 A031266 this_sequence A107573 A081308 A070210

Adjacent sequences: A109298 A109299 A109300 this_sequence A109302 A109303 A109304

nonn

Jon Awbrey (jawbrey(AT)att.net), Jun 24 2005 - Jul 08 2005

1,2

a(n) is also the sorted version of A057335 which is generated recursively using the formula A057335 = A057334 * A057335(repeated), where A057334 = A000040(A000120). - Alford Arnold (arnold1940(AT)aol.com), Nov 11 2001

Square-free kernels of these numbers are primorial numbers. See A080404. - Labos E. (labos(AT)ana.sote.hu), Mar 19 2003

If u and v are terms then so is u*v. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 24 2004

Except for the initial value a(0) = 1, a(n) gives the canonical primal code of the n-th finite sequence of positive integers, where n = (prime_1)^c_1 * ... * (prime_k)^c_k is the code for the finite sequence c_1, ..., c_k. See examples of primal codes at A106177. - Jon Awbrey (jawbrey(AT)att.net), Jun 22 2005

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..1001

Leroy Quet, Home Page (listed in lieu of email address)

60 is included because 60 = 2^2 * 3 * 5 and 2, 3 and 5 are consecutive primes beginning at 2.

Sequence A057335 begins

1..2..4..6..8..12..18..30..16..24..36..60..54..90..150..210... which is equal to

1..2..2..3..2...3...3...5...2...3...3...5...3...5....5....7... times

1..1..2..2..4...4...6...6...8...8..12..12..18..18...30...30...

Select[Range[1000], #==1||FactorInteger[ # ][[ -1, 1]]==Prime[Length[FactorInteger[ # ]]]&]

Cf. A057335, A056808, A025487, A007947, A002110, A080404, A106177.

Cf. A124829, A124830, A124831, A124833.

Sequence in context: A140110 A128397 A120383 this_sequence A140067 A067946 A145853

Adjacent sequences: A055929 A055930 A055931 this_sequence A055933 A055934 A055935

easy,nonn

Leroy Quet Jul 17 2000

0,3

a(n) is the sequence of first differences of A050924. Conversely, A050924 is the sequence of partial sums of a(n). This can be seen as follows. Let P(0) c P(1) c ... c P(n) c ... be an increasing sequence of sets of partial functions that is defined by the recursion: P(0) = {the empty function}, P(n+1) = {partial functions: P(n) -> P(n)}. Then |P(n)| = A050924(n+1) = number of positive integers of rote height at most n, hence |P(n)| - |P(n-1)| = a(n) = number of positive integers of rote height exactly n.

a(n) is defined by the recursion a(n+1) = (b(n) + 1)^b(n) - b(n), where a(0) = 1 and b(n) = Sum_[0, n] a(i).

Table of Rotes and Primal Functions for Positive Integers of Rote Height 2

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

o-o ` ` o-o ` ` ` o-o ` o-o o-o ` ` o-o o-o ` ` ` o-o o-o ` ` o-o o-o o-o

| ` ` ` | ` ` ` ` | ` ` | ` | ` ` ` | ` | ` ` ` ` | ` | ` ` ` | ` | ` | `

o-o ` o-o ` ` o-o o-o ` o---o ` ` o-o ` o-o ` o-o o---o ` ` o-o ` o---o `

| ` ` | ` ` ` | ` | ` ` | ` ` ` ` | ` ` | ` ` | ` | ` ` ` ` | ` ` | ` ` `

O ` ` O ` ` ` O===O ` ` O ` ` ` ` O=====O ` ` O===O ` ` ` ` O=====O ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

2:1 ` 1:2 ` ` 1:1 2:1 ` 2:2 ` ` ` 1:2 2:1 ` ` 1:1 2:2 ` ` ` 1:2 2:2 ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

3 ` ` 4 ` ` ` 6 ` ` ` ` 9 ` ` ` ` 12` ` ` ` ` 18` ` ` ` ` ` 36` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Cf. A007097, A050924, A061396, A062504, A062537, A062860.

Cf. A109301, A106177, A108352, A108371, A111791 to A111800.

Sequence in context: A076910 A116257 A075984 this_sequence A124272 A128820 A067485

Adjacent sequences: A109297 A109298 A109299 this_sequence A109301 A109302 A109303

nonn

Jon Awbrey (jawbrey(AT)att.net), Jul 04 2005, revised Sep 06 2005