A114913 -id:A114913 - cp's OEIS frontend

A114914 Terms in A114913 that are not in A111174.

76, 101, 149, 198, 201, 251, 326, 351, 368, 394, 426, 451, 476, 489, 492, 514, 601, 626, 639, 683, 688, 701, 801, 835, 879, 884, 933, 973, 976, 1051, 1076, 1098, 1168, 1176, 1178, 1201, 1215, 1227, 1251, 1301, 1351, 1359, 1374, 1376, 1459, 1551, 1570

Offset: 1

Christian G. Bower, Jan 06 2006

nonn

If 24*k+1 = 25*p for some prime p, then k will be in this sequence, implying that this sequence is infinite. - Dean Hickerson, Jan 19 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A111174, A114913.

q[n_] := Module[{f = FactorInteger[n], f1, f2}, f1 = Select[f, MemberQ[{1, 5, 7, 11}, Mod[First[#], 24]] &]; f2 = Select[f, MemberQ[{13, 17, 19, 23}, Mod[First[#], 24]] &]; AllTrue[f2[[;;, 2]], EvenQ] && Count[f1[[;;, 2]], ?OddQ] == 1]; Select[Range[1600], CompositeQ[24 * # + 1] && q[24 * # + 1] &] (* _Amiram Eldar, Aug 24 2024 *)

A113780 Number of solutions to 24n+1 = x^2+24y^2, x a positive integer, y an integer.

Original entry on oeis.org

1, 3, 3, 2, 2, 3, 4, 1, 2, 4, 2, 4, 1, 2, 2, 1, 8, 2, 2, 2, 0, 4, 1, 4, 2, 2, 5, 4, 2, 0, 4, 4, 2, 0, 0, 3, 4, 4, 4, 2, 3, 4, 2, 2, 4, 0, 0, 2, 2, 4, 2, 9, 2, 0, 2, 2, 4, 1, 4, 0, 4, 4, 2, 0, 4, 4, 4, 2, 0, 2, 1, 8, 0, 2, 2, 2, 6, 1, 2, 4, 0, 4, 4, 2, 2, 0, 8, 2, 2, 2, 2, 0, 1, 8, 0, 2, 4, 0, 0, 2, 5, 6, 4, 2, 4

Offset: 0

Christian G. Bower, Jan 20 2006, based on a message from Dean Hickerson

nonn

If 24*n+1 is not a square or if sqrt(24*n+1) == 1 or 11 (mod 12), then A000009(n) == a(n) (mod 4), otherwise A000009(n) == a(n) + 2 (mod 4).
Implied by the arithmetic of Q[sqrt(-6)]: Let 24*n+1 = p_1^e_1 * ... * p_r^e_r * q_1^f_1 * ... * q_s^f_s, where the p_i's are distinct primes == 1, 5, 7, or 11 (mod 24) and the q_i's are distinct primes == 13, 17, 19, or 23 (mod 24). If some f_i is odd, then a(n) = 0. Otherwise, a(n) = (e_1 + 1) * ... * (e_r + 1). a(n) == 2 (mod 4) iff all of the f_i's are even and all but one of the e_i's are even and the one e_i which is odd is == 1 (mod 4). Since A000009(n) and a(n) are both odd if 24*n+1 is a square, we can replace a by A000009 in this.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			If n=51, the solutions (x,y) are: (7,+-7), (19,+-6), (25,+-5), (29,+-4), (35,0) so a(51)=9.
G.f. = 1 + 3*x + 3*x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + x^7 + 2*x^8 + 4*x^9 + ...
G.f. = q + 3*q^25 + 3*q^49 + 2*q^73 + 2*q^97 + 3*q^121 + 4*q^145 + q^169 + 2*q^193 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001318 generalized pentagonal numbers, indices of odd values of a(n) and A000009.
Cf. A114913 = values k such that A000009(k) == 2 (mod 4) and such that a(k) == 2 (mod 4).
Cf. A128580, A129402, A134177, A190615.

a[ n_] := If[ n < 0, 0, With[{m = 24 n + 1}, Sum[ KroneckerSymbol[ -12, d] KroneckerSymbol[ 2, m/d], {d, Divisors @ m}]]]; (* Michael Somos, Jun 08 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 3, 0, x] / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jun 08 2013 *)

{a(n) = if( n<0, 0, n = 24*n + 1; sumdiv( n, d, kronecker( -12, d) * kronecker( 2, n/d)))}; /* Michael Somos, Mar 11 2007 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^3 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A)), n))}; /* Michael Somos, Jun 08 2012 */

Expansion of phi(x) * phi(-x^3) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Jun 08 2012
Expansion of f(x, x) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jun 08 2013
Expansion of eta(q^2)^6 * eta(q^3)^2 / (eta(q)^3 * eta(q^4)^2 * eta(q^6)) in powers of q. - Michael Somos, Jun 08 2012
Euler transform of period 12 sequence [ 3, -3, 1, -1, 3, -4, 3, -1, 1, -3, 3, -2, ...]. - Michael Somos, Jun 08 2012
a(n) = A128580(12*n) = A129402(12*n) = A134177(12*n) = A190615(12*n). - Michael Somos, Jun 08 2012

A114912 2^a(n) divides A000009(n) but 2^(a(n)+1) does not.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 0, 1, 3, 1, 2, 0, 1, 1, 0, 5, 1, 1, 1, 6, 2, 0, 3, 1, 1, 0, 6, 1, 8, 3, 2, 1, 6, 9, 0, 2, 3, 5, 1, 0, 2, 1, 1, 3, 11, 8, 1, 1, 6, 1, 0, 1, 10, 1, 1, 2, 0, 3, 6, 7, 2, 1, 9, 2, 3, 2, 1, 13, 1, 0, 5, 9, 1, 1, 1, 1, 0, 1, 3, 9, 2, 6, 1, 1, 6, 6, 1, 1, 1, 1, 11, 0, 5, 6, 1, 2, 8, 6, 1, 0, 1

Offset: 0

Christian G. Bower, Jan 06 2006

nonn

Almost all members of A000009 are divisible by 2^k for any k, therefore almost all a(n)>k for any k.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Krishnaswami Alladi, Partition Identities Involving Gaps and Weights, Transactions of the American Mathematical Society, Vol. 349, No. 12 (Dec 1997), pp. 5001-5019.
Basil Gordon and Ken Ono, Divisibility of certain partition functions by powers of primes, The Ramanujan Journal, Vol. 1, No. 1 (1997), pp. 25-34; alternative link.

Cf. A000009, A007814.

Cf. A001318 (positions of 0's), A114913 (positions of 1's), A115251 (least inverse).

a[n_] := IntegerExponent[PartitionsQ[n], 2]; Array[a, 100, 0] (* Amiram Eldar, Aug 24 2024 *)

a(n) = A007814(A000009(n)). - Max Alekseyev, Apr 27 2010

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A114914 Terms in A114913 that are not in A111174.

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A113780 Number of solutions to 24n+1 = x^2+24y^2, x a positive integer, y an integer.

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A114912 2^a(n) divides A000009(n) but 2^(a(n)+1) does not.

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cp's OEIS Frontend

A114914 Terms in A114913 that are not in A111174.

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A113780 Number of solutions to 24*n+1 = x^2+24*y^2, x a positive integer, y an integer.

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PARI

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A114912 2^a(n) divides A000009(n) but 2^(a(n)+1) does not.

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A113780 Number of solutions to 24n+1 = x^2+24y^2, x a positive integer, y an integer.