0,2

Number of ordered trees having nodes of outdegree 0,1,2 and such that all leaves are at level n. Example: a(2)=6 because, denoting by I a path of length 2 and by Y a Y-shaped tree with 3 edges, we have I, Y, I*I, I*Y, Y*I, Y*Y, where * denotes identification of the roots. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 31 2002

a(n) has at least n different prime factors [Saidak]

Subsequence of square-free numbers (A005117). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 15 2004

For prime factors see A007996.

Curtiss shows that if the reciprocal sum of the multiset S = {x_1, x_2, ..., x_n} is 1, then max(S) <= a(n). - Charles R Greathouse IV, Feb 28 2007

The number of reduced ZBDDs for Boolean functions of n variables in which there is no zero sink. (ZBDDs are "zero-suppressed binary decision diagrams.") For example, a(2)=6 because of the 2-variable functions whose truth tables are 1000, 1010, 1011, 1100, 1110, 1111. - D. E. Knuth, Jun 04 2007

Using the methods of Aho and Sloane, Fibonacci Quarterly 11 (1973), 429-437, it is easy to show that a(n) is the integer just a tiny bit below the real number theta^{2^n}-1/2, where theta =~ 1.597910218 is the exponential of the rapidly convergent series ln(3/2)+sum_{n >= 0} ln(1+(2a_n+1)^{-2}). For example, theta^64-1/2 =~ 3263442.0000000383. - D. E. Knuth, Jun 04 2007

a(n) = A139145(2^(n+1) - 1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 10 2008

a(n+1) = a(n) th oblong (or promic, pronic, or heteromecic) numbers (A002378). a(n+1) = A002378(a(n)) = A002378(a(n-1)) * (A002378(a(n-1)) + 1). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 13 2009]

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

D. R. Curtiss, "On Kellogg's Diophantine Problem". The American Mathematical Monthly Vol. 29, No. 10 (1922), pp. 380-387.

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 94.

M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see p. 577.

Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Dec 2006

A. Urquhart, The complexity of propositional proofs, Bull. Symbolic Logic, 1 (1995) pp. 425-, esp. p. 434.

N. J. A. Sloane, Table of n, a(n) for n = 0..12

Index entries for sequences of form a(n+1)=a(n)^2 + ...

a(n) = A000058(n)-1 = A000058(n-1)^2-A000058(n-1) = 1/(1-sum{j<n}[1/A000058(j)]) where A000058 is Sylvester's sequence. - Henry Bottomley (se16(AT)btinternet.com), Jul 23 2001

a(n) = floor(c^(2^n)) where c=1.597910218031873178338070118157... - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 06 2002

a(1)=1, a(n) = prod(k=1, n-1, a(k)+1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 13 2003

If an (additional) initial 1 is inserted, a(n) = sum_{k<n} a(k)^2. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 11 2009]

a=1; lst={}; Do[a=a^2+a; AppendTo[lst, a], {n, 0, 9}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 20 2009]

(PARI) a(n)=if(n<1, n>=0, a(n-1)+a(n-1)^2)

Cf. A003687.

Cf. A011782. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 11 2009]

Sequence in context: A123137 A014117 A054377 this_sequence A100016 A000610 A023363

Adjacent sequences: A007015 A007016 A007017 this_sequence A007019 A007020 A007021

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

The next term is too large to include.

1,2

"Somebody incorrectly remembered Fermat's little theorem as saying that the congruence a^{n+1} = a (mod n) holds for all a if n is prime" (Zagier). The sequence gives the set of integers n for which this property is in fact true.

If i = j (mod n), then m^i = m^j (mod n) for all m. The latter congruence generally holds for any (m, n)=1 with i = j (mod k), k being the order of m modulo n, i.e. the least power k for which m^k = 1 (mod n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 04 2002

J. Dyer-Bennet, "A Theorem in Partitions of the Set of Positive Integers", Amer. Math. Monthly, 47(1940) pp. 152-4.

D. Zagier, Problems posed at the St Andrews Colloquium, 1996

Sequence in context: A152479 A115961 A123137 this_sequence A054377 A007018 A100016

Adjacent sequences: A014114 A014115 A014116 this_sequence A014118 A014119 A014120

nonn,fini,full,nice

David Broadhurst (D.Broadhurst(AT)open.ac.uk)

0,2

If n=1, then the prime immediately greater than n is 2. Hence the next number is n*p = 1*2 = 2.

If n=2, then the next prime is 3, so the next number in the sequence is 2*3=6.

If n=6, then the next prime is 7, so the next number in the sequence is 6*7=42

a[0] = 1; a[n_] := a[n - 1]*NextPrime[a[n - 1]]; Table[ a[n], {n, 0, 9}] (from Robert G. Wilson v Nov 23 2004)

Similar to A074839

Sequence in context: A014117 A054377 A007018 this_sequence A000610 A023363 A091241

Adjacent sequences: A100013 A100014 A100015 this_sequence A100017 A100018 A100019

nonn

Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Nov 18 2004

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 23 2004