A365167 -id:A365167 - cp's OEIS frontend

A365126 Number of representations of n as the sum of a prime number and a fourth power of a nonnegative integer.

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

Offset: 1

Ilya Gutkovskiy, Aug 24 2023

nonn

Cf. A002471, A302354, A365127, A365166, A365167.

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

G.f.: (Sum_{i>=1} x^prime(i)) * (Sum_{j>=0} 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}]]

A365169 Numbers that are not the sum of a prime number and a fourth power of a positive integer.

Original entry on oeis.org

1, 2, 5, 7, 9, 10, 11, 13, 15, 16, 17, 22, 25, 26, 28, 31, 34, 36, 37, 40, 41, 43, 46, 49, 50, 51, 52, 55, 56, 58, 61, 64, 65, 66, 67, 70, 71, 73, 76, 78, 79, 81, 82, 85, 91, 93, 96, 97, 101, 103, 106, 107, 109, 111, 115, 116, 120, 121, 126, 127, 130, 131, 133, 135

Offset: 1

Ilya Gutkovskiy, Aug 24 2023

nonn

Cf. A064233, A211167, A365166, A365167, A365168.

nmax = 138; Position[CoefficientList[Series[Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^4, {j, 1, nmax^(1/4)}], {x, 0, nmax}], x] // Rest, 0] // Flatten

A365292 a(n) is the least positive integer that can be expressed as the sum of a prime number and a fourth power of a positive integer in exactly n ways.

Original entry on oeis.org

1, 3, 18, 644, 1298, 6662, 14948, 29102, 77579, 102044, 181838, 367022, 545828, 754652, 1095254, 1985807, 1875902, 2672339, 4102958, 4851593, 7444799, 6826874, 10715858, 16398674, 18805058, 23502134, 31500614, 34967159, 41824499, 47158793, 54659384, 67269347, 83885588

Offset: 0

Ilya Gutkovskiy, Aug 31 2023

nonn

			For n = 3: 644 = 19 + 5^4 = 563 + 3^4 = 643 + 1^4.

Cf. A000040, A000583, A064283, A365167, A365290, A365291.

More terms from Jinyuan Wang, Sep 13 2023

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A365126 Number of representations of n as the sum of a prime number and a fourth power of a nonnegative 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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A365169 Numbers that are not the sum of a prime number and a fourth power of a positive integer.

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A365292 a(n) is the least positive integer that can be expressed as the sum of a prime number and a fourth power of a positive integer in exactly n ways.

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