A365169 -id:A365169 - cp's OEIS frontend

A365167 Number of representations of n as the sum of a prime number and a fourth power of a positive integer.

0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 2, 1, 0, 0, 1, 0, 2, 0, 1, 1, 1, 0, 0, 1, 1, 1

Offset: 1

Ilya Gutkovskiy, Aug 24 2023

nonn

Cf. A064272, A283760, A365126, A365168, A365169.

nmax = 120; CoefficientList[Series[Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^4, {j, 1, nmax^(1/4)}], {x, 0, nmax}], x] // Rest

G.f.: (Sum_{i>=1} x^prime(i)) * (Sum_{j>=1} x^(j^4)).

A365168 Numbers that are the sum of a prime number and a fourth power of a positive integer.

Original entry on oeis.org

3, 4, 6, 8, 12, 14, 18, 19, 20, 21, 23, 24, 27, 29, 30, 32, 33, 35, 38, 39, 42, 44, 45, 47, 48, 53, 54, 57, 59, 60, 62, 63, 68, 69, 72, 74, 75, 77, 80, 83, 84, 86, 87, 88, 89, 90, 92, 94, 95, 98, 99, 100, 102, 104, 105, 108, 110, 112, 113, 114, 117, 118, 119, 122, 123, 124, 125

Offset: 1

Ilya Gutkovskiy, Aug 24 2023

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A058654, A307647, A365127, A365167, A365169.

N:= 200: # for terms up to N
P:= select(isprime,{$2..N-1}):
F:= {seq(i^4,i=1..floor(N^(1/4)))}:
sort(convert({seq(seq(s+t,s=P),t=F)},list)); # Robert Israel, Aug 28 2024

nmax = 125; f[x_] := Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^4, {j, 1, nmax^(1/4)}]; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

A365166 Numbers that are not the sum of a prime number and a fourth power of a nonnegative integer.

Original entry on oeis.org

1, 9, 10, 15, 16, 22, 25, 26, 28, 34, 36, 40, 46, 49, 50, 51, 52, 55, 56, 58, 64, 65, 66, 70, 76, 78, 81, 82, 85, 91, 93, 96, 106, 111, 115, 116, 120, 121, 126, 130, 133, 135, 136, 141, 144, 145, 146, 156, 159, 161, 162, 166, 169, 171, 172, 175, 176, 177, 185, 186, 187, 196

Offset: 1

Ilya Gutkovskiy, Aug 24 2023

nonn

Cf. A014090, A045911, A356295, A365126, A365127, A365169.

nmax = 197; Position[CoefficientList[Series[Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^4, {j, 0, nmax^(1/4)}], {x, 0, nmax}], x] // Rest, 0] // Flatten

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A365167 Number of representations of n as the sum of a prime number and a fourth power of a positive integer.

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A365168 Numbers that are the sum of a prime number and a fourth power of a positive integer.

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A365166 Numbers that are not the sum of a prime number and a fourth power of a nonnegative integer.

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