A258831 -id:A258831 - cp's OEIS frontend

A098098 a(n) = sigma(6*n+5)/6.

1, 2, 3, 4, 5, 8, 7, 8, 9, 10, 14, 12, 16, 14, 15, 20, 17, 18, 19, 24, 26, 22, 23, 28, 25, 32, 32, 28, 29, 30, 38, 32, 33, 40, 40, 44, 42, 38, 39, 40, 57, 42, 43, 44, 45, 62, 47, 56, 49, 56, 62, 52, 53, 60, 64, 68, 64, 58, 59, 60, 74, 72, 70, 64, 65, 80, 67, 76, 80, 70, 93, 72

Offset: 0

Vladeta Jovovic, Sep 14 2004

Euler transform of period 6 sequence [2, 0, 0, 0, 2, -4, ...].
Expansion of q^(-5/6) * (eta(q)^-1 * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q. - Michael Somos, Sep 16 2004
2*a(n) is the number of bipartitions of 2*n+1 that are 3-cores. See Baruah and Nath. - Michel Marcus, Apr 13 2020

			G.f. =1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 8*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 10*x^9 + ...
G.f. = q^5 + 2*q^11 + 3*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 7*q^41 + 8*q^47 + 9*q^53 + ...

Ivan Neretin, Table of n, a(n) for n = 0..10000
Nayandeep Deka Baruah and Kallol Nath, Infinite families of arithmetic identities and congruences for bipartitions with 3-cores, Journal of Number Theory, Volume 149, April 2015, Pages 92-104.

Cf. A000203, A016969, A033686, A097723, A121443, A125514, A133739, A134077, A134078, A134079, A186100, A232343, A232356, A252650, A238831.

Basis( ModularForms( Gamma0( 36), 2), 432)[6]; /* Michael Somos, Jul 09 2018 */

Table[DivisorSigma[1, 6 n + 5]/6, {n, 0, 71}] (* Ivan Neretin, Apr 30 2016 *)

PARI
```
a(n) = sigma(6*n + 5)/6
```

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A))^2, n))} /* Michael Somos, Sep 16 2004 */

G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)))^2. - Michael Somos, Sep 16 2004
From Michael Somos, Jul 09 2018: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252650. -
Convolution square of A121444.
A232343(2*n) = (-1)^n * A258831(n) = A000203(6*n + 4) = a(n). A033686(2*n) = -A134079(2*n + 1) = 2 * a(n). A121443(6*n + 5) = A133739(6*n + 5) =  A232356(6*n + 5) = A134077(3*n + 2) = 6 * a(n). A125514(6*n + 5) = 24 * a(n). A134078(6*n + 5) = -36 * a(n). A186100(6*n + 5) = -72 * a(n). (End)
From Amiram Eldar, Dec 16 2022: (Start)
a(n) = A000203(A016969(n))/6.
Sum_{k=1..n} a(k) = (Pi^2/18) * n^2 + O(n*log(n)). (End)

A121613 Expansion of psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 6, -8, 13, -12, 14, -24, 18, -20, 32, -24, 31, -40, 30, -32, 48, -48, 38, -56, 42, -44, 78, -48, 57, -72, 54, -72, 80, -60, 62, -104, 84, -68, 96, -72, 74, -124, 96, -80, 121, -84, 108, -120, 90, -112, 128, -120, 98, -156, 102, -104, 192, -108, 110

Offset: 0

Michael Somos, Aug 10 2006

sign

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

Number 33 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 4*x + 6*x^2 - 8*x^3 + 13*x^4 - 12*x^5 + 14*x^6 - 24*x^7 + ...
G.f. = q - 4*q^3 + 6*q^5 - 8*q^7 + 13*q^9 - 12*q^11 + 14*q^13 - 24*q^15 + ...

J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.179
Hardy, et al., Collected Papers of Srinivasa Ramanujan, p. 326, Question 359.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Y. 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. A001938, A008438, A097723, A112610, A134343, A258831, A258835.

A := Basis( ModularForms( Gamma0(16), 2), 110); A[2] - 4*A[4]; /* Michael Somos, Jun 10 2015 */

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^2 / (4 q^(1/2)), {q, 0, n}]]; (* Michael Somos, Jun 22 2012 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^4] / QPochhammer[ q^2])^4, {q, 0, n}]; (* Michael Somos, Oct 14 2013 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[1, 2 n + 1]]; (* Michael Somos, Jun 15 2015 *)

{a(n) = if( n<0, 0, (-1)^n * sigma(2*n + 1))};

A = ModularForms( Gamma0(16), 2, prec=110).basis(); A[1] - 4*A[3]; # Michael Somos, Jun 27 2013

Expansion of q^(-1/2) * (eta(q) * eta(q^4) / eta(q^2))^4 in powers of q.
Expansion of q^(-1/2)/4 * k * k' * (K / (Pi/2))^2 in powers of q where k, k', K are Jacobi elliptic functions. - Michael Somos, Jun 22 2012
Euler transform of period 4 sequence [ -4, 0, -4, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^n, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1 (mod 4), b(p^e) = (-1)^e * (p^(e+1) - 1) / (p - 1) if p == 3 (mod 4).
Given g.f. A(x), then B(x) = 4 * Integral_{0..x} A(x^2) dx = arcsin(4 * x * A001938(x^2)) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = sin(u + v) / 2 - sin((u - v) / 2). - Michael Somos, Oct 14 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 27 2013
G.f.: (Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)))^4.
G.f.: Sum_{k>0} -(-1)^k * (2*k - 1) * x^(k - 1) / (1 + x^(2*k - 1)).
G.f.: (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(4*k)))^4.
G.f.: (Sum_{k>0} (-1)^floor(k/2) * x^((k^2 - k)/2))^4.
G.f.: Sum_{k>0} (-1)^k * (2*k - 1) * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = (-1)^n * A008438(n). a(2*n) = A112610(n). a(2*n + 1) = -4 * A097723(n).
Convolution square of A134343. - Michael Somos, Jun 20 2012
a(3*n + 2) = 6 * A258831(n). a(4*n + 3) = -8 * A258835(n). - Michael Somos, Jun 11 2015

A258832 Expansion of psi(-x^3) * f(-x, x^2) in powers of x where psi(), f(,) are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jun 11 2015

sign

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

			G.f. = 1 - x + x^2 - x^3 + x^4 - 2*x^5 - x^7 + x^8 - x^9 + 2*x^10 + ...
G.f. = q^5 - q^17 + q^29 - q^41 + q^53 - 2*q^65 - q^89 + q^101 - q^113 + ...

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. A121444, A258831.

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 12 n + 5, KroneckerSymbol[ -4, #] &] / 2];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^2 QPochhammer[ x, -x] / QPochhammer[ x^3, -x^3], {x, 0, n}];

{a(n) = if( n<0, 0, (-1)^n * sumdiv( 12*n + 5, d, kronecker( -4, d)) / 2)};

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

Expansion of q^(-5/12) * eta(q) * eta(q^4) * eta(q^6)^4 / (eta(q^2)^2 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -1, 1, 0, 0, -1, -2, -1, 0, 0, 1, -1, -2, ...].
G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(3*k))^2 * (1 - x^k + x^(2*k))^3 / (1 - x^(2*k) + x^(4*k)).
a(n) = (-1)^n * A121444(n). Convolution square is A258831.

A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

Original entry on oeis.org

1, 2, 4, 4, 5, 6, 7, 10, 9, 12, 11, 14, 16, 14, 15, 16, 20, 22, 19, 20, 21, 22, 31, 28, 28, 26, 30, 34, 29, 30, 36, 32, 40, 38, 35, 36, 37, 56, 39, 40, 41, 42, 52, 48, 57, 50, 47, 62, 49, 50, 56, 60, 64, 54, 55, 62, 57, 70, 68, 60, 66, 62, 76, 70, 70, 76

Offset: 1

M. F. Hasler, Dec 08 2017

Robert G. Wilson v observes in A280098 that {1, 3, 4, 6, 8, 12, 24} seem to be the only positive integers k such that sigma(kn-1)/k is an integer for all n > 0.

Muniru A Asiru, Table of n, a(n) for n = 1..100000

Cf. A280098 (analog for k = 24), A097723 (analog for k = 4), A033686 (analog for k = 3), A000203 (sigma, also the  analog for k = 1).
The analog for k = 8 is A258835, up to the offset.
The analog for k = 6 is A098098 (up to the offset), a signed variant of this and the preceding one is A258831.
Cf. A086463.

sequence := List([1..10^5], n-> Sigma(12 *n-1)/12); # Muniru A Asiru, Dec 28 2017

with(numtheory):
seq(sigma(12*n-1)/12, n=1..10^3); # Muniru A Asiru, Dec 28 2017

Array[DivisorSigma[1, 12 # - 1]/12 &, 66] (* Michael De Vlieger, Dec 08 2017 *)

PARI
```
vector(90,n,sigma(12*n-1)/12)
```

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Mar 28 2024

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A098098 a(n) = sigma(6*n+5)/6.

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A121613 Expansion of psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.

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A258832 Expansion of psi(-x^3) * f(-x, x^2) in powers of x where psi(), f(,) are Ramanujan theta functions.

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A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

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