A113605 -id:A113605 - cp's OEIS frontend

A113924 a(n) = gcd(A113605(n+1), A113605(n)). Also, for n >= 2, a(n) = A113605(n+2) - A113605(n-1).

1, 1, 1, 2, 1, 3, 1, 1, 4, 1, 1, 5, 1, 2, 1, 1, 1, 1, 8, 1, 3, 1, 1, 2, 1, 1, 1, 11, 2, 1, 1, 1, 1, 2, 1, 1, 5, 1, 4, 1, 1, 1, 1, 2, 19, 1, 1, 1, 10, 1, 7, 3, 1, 2, 1, 1, 1, 1, 6, 1, 1, 1, 5, 6, 1, 1, 1, 1, 2, 1, 3, 7, 1, 4, 1, 1, 1, 5, 2, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Leroy Quet, Jan 30 2006

nonn

Cf. A113605.

A113605 := proc(n) option remember ; if n <=3 then 1 ; else A113605(n-3)+gcd(A113605(n-1),A113605(n-2)) ; fi ; end: A113924 := proc(n) gcd(A113605(n+1),A113605(n)) ; end; seq(A113924(n),n=1..80) ; # R. J. Mathar, Jan 31 2008

lista(nn) = my(w, x=1, y=1, z=1); print1("1"); for(n=2, nn, w=x+gcd(y, z); print1(", ", w-x); x=y; y=z; z=w); \\ Jinyuan Wang, Aug 10 2021

More terms from R. J. Mathar, Jan 31 2008

More terms from Jinyuan Wang, Aug 10 2021

A129449 Expansion of psi(-x) * psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 0, -2, 1, 0, 2, 0, 0, -2, 2, 0, 1, -1, 0, -2, 0, 0, 2, -2, 0, -2, 0, 0, 3, 0, 0, 0, 2, 0, 2, -2, 0, -2, 0, 0, 2, -1, 0, -2, 1, 0, 0, 0, 0, -4, 2, 0, 2, 0, 0, -2, 0, 0, 2, -2, 0, 0, 2, 0, 1, 0, 0, -2, 2, 0, 4, 0, 0, -2, 0, 0, 0, -3, 0, -2, 0, 0, 2, 0, 0, -2, 0, 0, 3, -2, 0, -2, 0, 0, 2, -2, 0, 0, 2, 0, 2, 0, 0, -2, 2, 0, 0, 0, 0

Offset: 0

Michael Somos, Apr 16 2007

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number 53 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - x - 2*x^3 + x^4 + 2*x^6 - 2*x^9 + 2*x^10 + x^12 - x^13 - 2*x^15 + ...
G.f. = q - q^3 - 2*q^7 + q^9 + 2*q^13 - 2*q^19 + 2*q^21 + q^25 - q^27 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033762, A112604, A113605, A112606, A112608, A112609, A121361, A123884, A129451.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)] / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := With[ {m = 2 n + 1}, If[ m < 1, 0, Sum[ KroneckerSymbol[ 12, d] KroneckerSymbol[ -4, m/d], {d, Divisors[ m]}]]]; (* Michael Somos, Jul 09 2015 *)

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -4, d) * kronecker( 12, n/d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)/ (eta(x^2 + A) * eta(x^6 + A)), n))};

Expansion of q^(-1/2) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, -2, -1, -1, 0, -1, -1, -2, 0, -1, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5, 11 (mod 12), b(p^e) = e+1 if p == 1 (mod 12), b(p^e) = (-1)^e * (e+1) if p == 7 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A033762(n). a(2*n) = A112604(n). a(2*n + 1) = -A112605(n). a(3*n) = A129451(n). a(3*n + 1) = -a(n). a(3*n + 2) = 0.
a(4*n) = A112606(n). a(4*n + 1) = - A112608(n). a(4*n + 2) = 2 * A112607(n). a(4*n + 3) = - 2 * A112609(n).
a(6*n) = A123884(n). a(6*n + 3) = -2 * A121361(n).

A113604 Define f(k) = k + sum of digits of n. a(n) is the first prime that results by applying f zero or more times to n, or 0 if no such prime exists.

Original entry on oeis.org

2, 2, 3, 23, 5, 0, 7, 23, 0, 11, 11, 0, 13, 19, 0, 23, 17, 0, 19, 41, 0, 41, 23, 0, 37, 41, 0, 101, 29, 0, 31, 37, 0, 41, 43, 0, 37, 101, 0, 59, 41, 0, 43, 59, 0, 67, 47, 0, 101, 89, 0, 59, 53, 0, 89, 67, 0, 71, 59, 0, 61, 101, 0, 127, 89, 0, 67, 103, 0, 101, 71, 0, 73, 127, 0, 89

Offset: 1

Amarnath Murthy, Nov 09 2005

a(n) = n for n prime; a(n) = 0 for n > 3 and n == 0 (mod 3).

Conjecture: a(n) = 0 only if n == 0 (mod 3).

			a(4) = 23 since f(4) = 4+4 = 8, f(8) = 8+8 = 16, f(16) = 16+1+6 = 23.

Cf. A113605.

f(n) = local(k,s,d);k=n;s=0;while(k>0,d=divrem(k,10);k=d[1];s=s+d[2]);s+n {m=76;for(n=1,m,if(n>3&&n%3==0,print1(0,","),k=n;z=1000*n;while(k

Edited and extended by Klaus Brockhaus, Nov 10 2005

cp's OEIS Frontend

A113924 a(n) = gcd(A113605(n+1), A113605(n)). Also, for n >= 2, a(n) = A113605(n+2) - A113605(n-1).

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A129449 Expansion of psi(-x) * psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

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A113604 Define f(k) = k + sum of digits of n. a(n) is the first prime that results by applying f zero or more times to n, or 0 if no such prime exists.

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