A280245 -id:A280245 - cp's OEIS frontend

A298436 Expansion of Product_{k>=1} 1/(1 - x^prime(k))^2.

1, 0, 2, 2, 3, 6, 7, 12, 15, 22, 30, 40, 54, 72, 93, 122, 157, 202, 256, 326, 409, 512, 640, 792, 981, 1204, 1479, 1802, 2196, 2662, 3218, 3880, 4660, 5588, 6677, 7960, 9471, 11232, 13299, 15710, 18514, 21784, 25570, 29968, 35047, 40922, 47698, 55500, 64480, 74786, 86618

Offset: 0

Ilya Gutkovskiy, Jan 19 2018

nonn

Number of partitions of n into prime parts of 2 kinds.

Self-convolution of A000607.

			a(5) = 6 because we have [5a], [5b], [3a, 2a], [3a, 2b], [3b, 2a] and [3b, 2b].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Index entries for related partition-counting sequences

Cf. A000040, A000607, A000712, A280245.

nmax = 50; CoefficientList[Series[Product[1/(1 - x^Prime[k])^2, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^prime(k))^2.

log(a(n)) ~ 2*Pi*sqrt(2*n/(3*log(n/2))). - Vaclav Kotesovec, Jan 12 2021

A369789 Number of different coefficient values in expansion of Product_{k=1..n} (1+x^prime(k))^2.

Original entry on oeis.org

1, 3, 4, 5, 10, 20, 29, 47, 69, 92, 122, 153, 190, 231, 274, 321, 374, 433, 494, 561, 632, 705, 784, 867, 956, 1053, 1154, 1257, 1364, 1473, 1586, 1713, 1844, 1981, 2120, 2269, 2420, 2577, 2740, 2907, 3080, 3259, 3440, 3631, 3824, 4021, 4220, 4431, 4654, 4881, 5110

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Cf. A280245, A369788.

a(n) = #Set(Vec(prod(k=1, n, (1+x^prime(k))^2)));

from collections import Counter
from sympy import prime
def A369789(n):
    c = {0:1}
    for k in range(1,n+1):
        m, d = prime(k), Counter(c)
        for j in c:
            a = c[j]
            d[j+m] += a<<1
            d[j+(m<<1)] += a
        c = d
    return len(set(c.values()))+int(max(c)+1>len(c)) # Chai Wah Wu, Feb 01 2024

A298948 Expansion of Product_{k>=1} (1 - x^prime(k))^2.

Original entry on oeis.org

1, 0, -2, -2, 1, 2, 1, 0, 2, 2, -2, -6, -2, 2, 2, 0, 3, 2, -1, -6, -2, 2, 3, -2, 4, 6, 0, -10, -4, 0, 4, -2, 5, 8, 6, -12, -6, -4, -1, -6, 12, 10, 8, -12, -4, -4, 1, -18, 11, 18, 15, -20, -2, -8, 7, -18, 8, 12, 29, -24, 2, -8, 3, -34, 21, 6, 29, -32, 5, -8, 31, -52

Offset: 0

Seiichi Manyama, Jan 30 2018

sign

Self-convolution of A046675.

Cf. A046675, A280245, A298436.

cp's OEIS Frontend

A298436 Expansion of Product_{k>=1} 1/(1 - x^prime(k))^2.

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A369789 Number of different coefficient values in expansion of Product_{k=1..n} (1+x^prime(k))^2.

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A298948 Expansion of Product_{k>=1} (1 - x^prime(k))^2.

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