A261328 -id:A261328 - cp's OEIS frontend

A262930 Expansion of (psi(-q) / f(q^3))^2 in powers of q where psi(), f() are Ramanujan theta functions.

1, -2, 1, -4, 6, -2, 12, -16, 5, -28, 36, -12, 60, -76, 24, -120, 150, -46, 228, -280, 86, -416, 504, -152, 732, -878, 262, -1252, 1488, -442, 2088, -2464, 725, -3408, 3996, -1168, 5460, -6364, 1852, -8600, 9972, -2886, 13344, -15400, 4436, -20424, 23472

Offset: 0

Michael Somos, Oct 04 2015

sign

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

			G.f. = 1 - 2*x + x^2 - 4*x^3 + 6*x^4 - 2*x^5 + 12*x^6 - 16*x^7 + 5*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A139136, A261325, A261328, A261369, A263528, A263538.

a[ n_] := SeriesCoefficient[ (1/2) q^(-1/4) (EllipticTheta[ 2, Pi/4, q^(1/2)] / QPochhammer[ -q^3])^2, {q, 0, n}];

{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)^3 ))^2, n))};

Expansion of ( eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / ( eta(q^2) * eta(q^6)^3 ))^2 in powers of q.
Euler transform of period 12 sequence [ -2, 0, -4, -2, -2, 4, -2, -2, -4, 0, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A261369.
a(3*n) = A261320(n). a(3*n + 1) = -2 * A261325(n). a(3*n + 2) = A261369(n).
Convolution square of A139136.
a(2*n) = A263538(n). a(2*n + 1) = -2 * A263528(n).

A261296 Smaller of pairs (m, n), such that the difference of their squares is a cube and the difference of their cubes is a square.

Original entry on oeis.org

6, 384, 4374, 5687, 24576, 17576, 27783, 64350, 93750, 354375, 279936, 113750, 363968, 166972, 370656, 705894, 263736, 1572864, 1124864, 1778112, 3187744, 4225760, 4118400, 3795000, 3188646, 4145823, 4697550, 1111158, 730575, 6000000, 8171316, 2413071, 8573750

Offset: 1

Pieter Post, Aug 14 2015

nonn

The numbers come in pairs: (6,10), (384, 640) etc. The larger numbers of the pairs can be found in A261328. The sequence has infinite subsequences: Once a pair is in the sequence all its zenzicubic multiples (i.e., by a 6th power) are also in this sequence. Primitive solutions are (6,10), (5687, 8954), (27883, 55566), (64350, 70434), ....

Assumes m, n > 0 as otherwise (k^6, 0) will be a solution. Sequence sorted by increasing order of largest number in pair (A261328). - Chai Wah Wu, Aug 17 2015

			10^3 - 6^3 = 784 = 28^2, 10^2 - 6^2 = 64 = 4^3.
8954^3 - 5687^3 = 730719^2, 8954^2 - 5687^2 = 363^3.

H. E. Dudeney, 536 Puzzles & Curious Problems, Charles Scribner's Sons, New York, 1967, pp 56, 268, #177

Chai Wah Wu, Table of n, a(n) for n = 1..302
Gianlino, in reply to Smci, Solution method for "integers with the difference between their cubes is a square, and v.v.", Yahoo! answers, 2011

Cf. A000290, A000578, A001014, A261328.

def cube(z, p):
    iscube=False
    y=int(pow(z, 1/p)+0.01)
    if y**p==z:
        iscube=True
    return iscube
for n in range (1, 10**5):
    for m in range(n+1, 10**5):
        a=(m-n)*(m**2+m*n+n**2)
        b=(m-n)*(m+n)
        if cube(a, 2)==True and cube(b, 3)==True:
            print (n, m)

Added a(6) and more terms from Chai Wah Wu, Aug 17 2015

cp's OEIS Frontend

A262930 Expansion of (psi(-q) / f(q^3))^2 in powers of q where psi(), f() are Ramanujan theta functions.

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