A046675 -id:A046675 - cp's OEIS frontend

A367843 Maximum of the absolute value of the coefficients of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 6, 8, 12, 18, 30, 46, 70, 113, 186, 314, 531, 894, 1561, 2705, 4817, 8514, 15030, 26502, 47200, 84698, 151809, 273961, 496807, 900596, 1643185, 2999837, 5498916, 10111429, 18596096, 34306158, 63585519, 118215700

Offset: 0

Ilya Gutkovskiy, Feb 06 2024

nonn

Cf. A046675, A160089, A350457, A350514.

Table[Max[Abs[CoefficientList[Product[(1 - x^Prime[k]), {k, 1, n}], x]]], {n, 0, 46}]

from collections import Counter
from sympy import prime
def A367843(n):
    c = {0:1}
    for k in range(1,n+1):
        p, b = prime(k), Counter(c)
        for j in c:
            b[j+p] -= c[j]
        c = b
    return max(map(abs,c.values())) # Chai Wah Wu, Feb 06 2024

A379976 Absolute value of the minimum coefficient of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 4, 6, 8, 12, 18, 30, 46, 70, 113, 186, 314, 531, 894, 1561, 2705, 4817, 8502, 15030, 26502, 47200, 84698, 151809, 273961, 496807, 900596, 1643185, 2999067, 5498916, 10110030, 18596096, 34300223, 63585519, 118208807, 219235308, 405259618, 752027569, 1400505025

Offset: 0

Ilya Gutkovskiy, Jan 07 2025

nonn

Robert Israel, Table of n, a(n) for n = 0..1500

Cf. A046675, A086394, A350457, A350514, A367843.

P:= 1: R:= 1:
for i from 1 to 100 do
  P:= P * (1 - x^ithprime(i));
  R:= R, abs(min(coeffs(expand(P),x)))
od:
R; # Robert Israel, Feb 07 2025

Table[Min[CoefficientList[Product[(1 - x^Prime[k]), {k, 1, n}], x]], {n, 0, 50}] // Abs

a(n) = abs(vecmin(Vec(prod(k=1, n, 1-x^prime(k))))); \\ Michel Marcus, Jan 18 2025

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.

A300521 Expansion of Product_{k>=1} (1 - x^prime(k))^prime(k).

Original entry on oeis.org

1, 0, -2, -3, 1, 1, 3, 0, 9, 8, 4, -31, -12, -13, 20, -13, 48, -17, 74, -87, 8, -143, 175, -174, 349, -164, 369, -651, 520, -1004, 1142, -1218, 1652, -1739, 3291, -3933, 3546, -5743, 6170, -8022, 11435, -13230, 17196, -18706, 22958, -31884, 38420, -49802, 58916

Offset: 0

Ilya Gutkovskiy, Mar 08 2018

sign

Cf. A000040, A000607, A005063, A007441, A030009, A046675, A073592, A219224, A291647, A291696.

nmax = 48; CoefficientList[Series[Product[(1 - x^Prime[k])^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[-Sum[DivisorSum[k, Boole[PrimeQ[#]] #^2 &] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^A000040(k))^A000040(k).

G.f.: exp(-Sum_{k>=1} A005063(k)*x^k/k).

cp's OEIS Frontend

A367843 Maximum of the absolute value of the coefficients of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(n)).

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A379976 Absolute value of the minimum coefficient of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(n)).

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

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A300521 Expansion of Product_{k>=1} (1 - x^prime(k))^prime(k).

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