0,2

Also a(n)-1 is number of 1's in truth table for lexicographically least de Bruijn cycle (Fredricksen).

N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.

H. Fredricksen, The lexicographically least de Bruijn cycle, J. Combin. Theory, 9 (1970) 1-5.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, pp. 120, 172.

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (Jun 10, 1976), 459-467.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.

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

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).

R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

T. D. Noe, Table of n, a(n) for n = 0..200

Joerg Arndt, Fxtbook

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 18, 64

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 2

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 130

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

N. J. A. Sloane, On single-deletion-correcting codes

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Wolfram Research, Number of necklaces

Index entries for "core" sequences

Index entries for sequences related to necklaces

a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n/d).

For n=3 and n=4 the necklaces are {000,001,011,111} and {0000,0001,0011,0101,0111,1111}.

The analogous shift register sequences are {000..., 001001..., 011011..., 111...} and {000..., 00010001..., 00110011..., 0101..., 01110111..., 111... }.

with(numtheory); A000031 := proc(n) local d, s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+phi(d)*2^(n/d); od; RETURN(s/n); fi; end; [ seq(A000031(n), n=0..50) ];

a[n_] := Fold[ # 1 + EulerPhi[ # 2]2^(n/ # 2) &, 0, Divisors[n]]/n

(PARI) {A000031(n)=if(n==0, 1, sumdiv(n, d, eulerphi(d)*2^(n/d))/n)}. - Randall L. Rathbun, Jan 11 2002

Cf. A001037 (primitive solutions to same problem), A014580, A000016, A000013, A000029 (if turning over is allowed), A000011, A001371, A058766.

Rows sums of triangle in A047996.

Dividing by 2 gives A053634.

A008965(n) = a(n) - 1 allowing different offsets.

Cf. A008965, A053635, A052823.

Sequence in context: A018137 A084239 A049708 this_sequence A111023 A008324 A084074

Adjacent sequences: A000028 A000029 A000030 this_sequence A000032 A000033 A000034

nonn,easy,nice,core

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

There is an error in Fig. M3860 in the 1995 Encyclopedia of Integer Sequences: in the third line, the formula for A000031 = M0564 should be (1/n) sum phi(d) 2^(n/d).

1,1

T. D. Noe, Table of n, a(n) for n=1..200

N. Kitchloo and L. Pachter, An interesting result about subset sums (pdf)

a(n) = 1/n * sum_{d divides n and d is odd} 2^(n/d) * phi(d).

Table[a = Select[ Divisors[n], OddQ[ # ] &]; Apply[Plus, 2^(n/a)*EulerPhi[a]]/n, {n, 1, 35}]

Equals 2*A000016(n). The super-diagonal of A068009. Cf. also A000010, A000013, A051293, A053633. For odd n a(n) = A000031(n) (necklaces).

Cf. A053636, A054539, A082550.

Sequence in context: A022476 A000013 A064484 this_sequence A118406 A072488 A074818

Adjacent sequences: A063773 A063774 A063775 this_sequence A063777 A063778 A063779

nonn,nice

Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 16 2001

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 20 2001

1,1

a(n) = A159069(n) +1.

Cf. A159069, A159553.

Sequence in context: A007497 A126882 A006549 this_sequence A101128 A119345 A086391

Adjacent sequences: A159551 A159552 A159553 this_sequence A159555 A159556 A159557

nonn

Leroy Quet (q1qq2qqq3qqqq(AT)yahoo.com), Apr 14 2009

Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jun 18 2009

0,7

(PARI) {A068038(n)=local(v, v1); v=vector(17); v[1]=1; for(i=1, n, v1=vector(17); for(j=0, 16, v1[j+1]=v[j+1]+v[(j-i)%17+1]); v=v1); v[1]} (Max Alekseyev (maxale(AT)gmail.com), Jul 23 2005))

17th row of A068009.

Sequence in context: A015631 A116686 A135350 this_sequence A090741 A032234 A032255

Adjacent sequences: A068035 A068036 A068037 this_sequence A068039 A068040 A068041

nonn

Antti Karttunen, Feb 11 2002

Rechecked by Max Alekseyev, maxale(AT)gmail.com, Jul 23 2005

1,2

2^n and 2^n + 1 seem to be in the sequence for all n. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 04 2006

Cf. A109849.

Sequence in context: A044051 A083132 A118956 this_sequence A008749 A029000 A042962

Adjacent sequences: A109847 A109848 A109849 this_sequence A109851 A109852 A109853

nonn

Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 06 2005

More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 04 2006

1,1

Summing the ordered gaussian polynomial templates in an orthogonal manner yields the table A047969.

A105552 is constructed by summing values from ordered gaussian polynomials, therefore the present begins with one value each from the following diagonal sequences and sums to A047970(11).

A000124 A000330 A086602 A089574 A107600 A107601 A109125 A109126 A109820 A108538 A109821 A110553 ...

for example A110554(6) = A107601(6) = 7936

Cf. A047969, A047970, A110553.

Sequence in context: A136547 A158481 A158487 this_sequence A005912 A104677 A156375

Adjacent sequences: A110551 A110552 A110553 this_sequence A110555 A110556 A110557

fini,nonn,uned

Alford Arnold (Alford1940(AT)aol.com), Jul 30 2005

0,2

XOR is A003987.

Index entries for sequences defined by congruent products under XOR

Cf. A115834 shows this sequence in binary.

Sequence in context: A109840 A108293 A063339 this_sequence A052158 A095672 A073650

Adjacent sequences: A115830 A115831 A115832 this_sequence A115834 A115835 A115836

nonn

Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Feb 01 2006

0,3

In general, the number of p-wild partitions of n is equal to the number of partitions of n if and only if n<p. From n=p onward, there are many more p-wild partitions.

David P. Roberts, Wild Partitions and Number Theory Journal of Integer Sequences, Volume 10, Issue 6, Article 07.6.6, (2007)

The generating function is prod_{j=0}^infinity theta_2(2^(2^j-1) x)^(2^j) where theta_2(y) is the generating function for 2-cores A010054

a(2) = 4, since there are four quadratic algebras over Q_2 up to geometric equivalence, namely Q_2 times Q_2, Q_2(sqrt{-1}), Q_2(sqrt{2}) and Q_2(sqrt{-2})

Cf. A000041, A010054, A131140.

Sequence in context: A013468 A041907 A151450 this_sequence A152291 A041557 A123304

Adjacent sequences: A131136 A131137 A131138 this_sequence A131140 A131141 A131142

nonn

David P. Roberts (roberts(AT)morris.umn.edu), Jun 19 2007

3,1

a(n) >= m(n), where m(n) is the smallest integer of the same parity as n, which is >= (sum_{k=1..n^2} prime(k+1))/n. For example, sum_{k=1..5^2} prime(k+1)/5=231.8, so m(5)=233. Conjecture: for n>4, a(n)=m(n) or a(n)=m(n)+2.

Harvey Heinz, Prime magic squares

A. Lelechenko and N. Makarova, Examples of prime magic n X n squares with minimal magic constant for n=5..13.

N. Makarova, Smallest prime magic squares, Part I (in Russian)

N. Makarova, Smallest prime magic squares, Part II (in Russian)

Mathworld, Prime magic squares

PlanetMath, Prime magic squares

Stefano Tognon, Prime Analysis

From N. Makarova (natalimak1(AT)yandex.ru), Sep 26 2009 (Start)

Here is a 14 X 14 example:

[3 43 59 131 181 271 383 599 797 919 971 1039 1123 1193

1151 433 967 211 337 491 397 691 83 523 593 773 449 613

263 373 101 1063 877 617 419 911 787 241 151 839 739 331

503 439 809 1051 1091 659 157 1031 71 139 379 179 743 461

173 647 1069 389 1049 19 311 223 317 1103 283 947 499 683

547 13 1061 353 229 853 677 751 571 983 1201 29 193 251

643 269 887 733 23 409 1129 191 769 401 47 1109 149 953

163 881 673 107 431 487 991 631 829 109 349 367 811 883

1163 827 607 1171 443 653 463 5 457 577 31 293 601 421

509 1097 313 757 167 709 761 347 857 137 619 233 89 1117

1093 1019 7 521 1033 61 73 941 1009 859 701 11 127 257

53 467 97 307 1153 557 1021 569 359 937 821 113 977 281

907 17 823 641 661 929 67 719 79 587 479 563 1013 227

541 1187 239 277 37 997 863 103 727 197 1087 1217 199 41] (End)

Comment from N. J. A. Sloane, Sep 28 2009: this contains 192 consecutive primes, 3 to 1171, plus 1187, 1193, 1201, 1217.

For the 3 X 3 case see A024351. For the 4 X 4 magic square see the Mathworld link.

Cf. A073502, A073350, A125007.

nonn,more,new

Andrew Lelechenko (andrew.lelechenko(AT)gmail.com), Aug 28 2009 and Natalia Makarova (natalimak1(AT)yandex.ru), Sep 08 2009

Partially reworded by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 31 2009

Edited by N. J. A. Sloane, Sep 14 2009

a(11)-a(14) from N. Makarova, Sep 18 2009

Edited by Max Alekseyev (maxale(AT)gmail.com), Sep 22 2009

a(15)-a(31) from N. Makarova and S. Tognon. - Max Alekseyev (maxale(AT)gmail.com), Jan 28 2010

1,2

R. E. Krichevsky, Szhatie i poisk informatsii (Compressing and searching for information), Moscow, 1988, ISBN 5-256-00325-9.

The code for n is found as follows: from right to left, the truncated (without the leading 1) binary representations of n, int(log2(n)), int(log2(int(log2(n)))), etc., are written as long as they consist of at least one bit; then we write a 0 followed by log*(n) 1's.

Sequence in context: A072483 A081539 A141273 this_sequence A103761 A078755 A086285

Adjacent sequences: A010094 A010095 A010096 this_sequence A010098 A010099 A010100

nonn

Leonid Broukhis (leob(AT)mailcom.com)