A365126 -id:A365126 - 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)).

A365291 a(n) is the least positive integer that can be expressed as the sum of a prime number and a fourth power of a nonnegative integer in exactly n ways.

Original entry on oeis.org

1, 2, 3, 83, 1298, 4259, 11087, 22637, 65579, 102044, 181838, 234527, 467627, 754652, 991889, 1431113, 1875902, 2271407, 4102958, 4851593, 8458148, 6826874, 10715858, 15453983, 18805058, 23502134, 27671573, 34425893, 34967159, 47158793, 54659384, 78188129, 67269347

Offset: 0

Ilya Gutkovskiy, Aug 31 2023

nonn

			For n = 3: 83 = 2 + 3^4 = 67 + 2^4 = 83 + 0^4.

Cf. A000040, A000583, A365126, A365288, A365289, A365292.

More terms from Jinyuan Wang, Sep 13 2023

A365127 Numbers that are the sum of a prime number and a fourth power of a nonnegative integer.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 19, 20, 21, 23, 24, 27, 29, 30, 31, 32, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 53, 54, 57, 59, 60, 61, 62, 63, 67, 68, 69, 71, 72, 73, 74, 75, 77, 79, 80, 83, 84, 86, 87, 88, 89, 90, 92, 94, 95, 97, 98, 99, 100

Offset: 1

Ilya Gutkovskiy, Aug 24 2023

nonn

Cf. A014089, A307646, A365126, A365166, A365168.

nmax = 100; f[x_] := Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^4, {j, 0, 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

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.

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A365291 a(n) is the least positive integer that can be expressed as the sum of a prime number and a fourth power of a nonnegative integer in exactly n ways.

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A365127 Numbers that are the sum of a prime number and a fourth power of a nonnegative integer.

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Mathematica

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

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