A283760 -id:A283760 - cp's OEIS frontend

A302354 Expansion of (Sum_{i>=1} x^prime(i))*(Sum_{j>=0} x^(j^3)).

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

Offset: 1

Ilya Gutkovskiy, Apr 06 2018

nonn

Number of representations of n as the sum of a prime number and a nonnegative cube.

			a(11) = 2 because 11 = 3 + 2^3 = 11 + 0^3.

Cf. A002471, A010051, A010057, A045911, A064272, A283760.

nmax = 120; Rest[CoefficientList[Series[Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^3, {j, 0, nmax}], {x, 0, nmax}], x]]

G.f.: (Sum_{i>=1} x^prime(i))*(Sum_{j>=0} x^(j^3)).

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

Original entry on oeis.org

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)).

A365290 a(n) is the least positive integer that can be expressed as the sum of a prime number and a positive cube in exactly n ways.

Original entry on oeis.org

1, 3, 30, 128, 1130, 2214, 6654, 10358, 24496, 37599, 64034, 59455, 85377, 158435, 240074, 253628, 313407, 405925, 548802, 891845, 809384, 1317788, 1547004, 2049122, 1838349, 2516848, 3192927, 2448059, 4349417, 4709007, 4438311, 6483753, 6175237, 8306209

Offset: 0

Ilya Gutkovskiy, Aug 31 2023

nonn

			For n = 3: 128 = 3 + 5^3 = 101 + 3^3 = 127 + 1^3.

Rémy Sigrist, C++ program

Cf. A000040, A000578, A064283, A283760, A365289, A365292.

More terms from Rémy Sigrist, Sep 07 2023

A307647 Numbers that are the sum of a prime number and a positive cube.

Original entry on oeis.org

3, 4, 6, 8, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 24, 25, 27, 29, 30, 31, 32, 34, 37, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 54, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 77, 79, 80, 81, 83, 84, 86, 87, 88, 90, 91, 93, 94, 95, 97, 98, 100, 101, 102, 104, 105

Offset: 1

Ilya Gutkovskiy, Apr 19 2019

nonn

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

Cf. A000040, A000578, A058654, A211167, A283760, A307646.

N:= 200: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N-1,2)]):
C:= [seq(j^3,j=1..floor((N-2)^(1/3)))]:
sort(convert(select(`<=`,{seq(seq(p+c,p=P),c=C)},N),list)); # Robert Israel, Apr 22 2019

Exponents in expansion of (Sum_{i>=1} x^prime(i)) * (Sum_{j>=1} x^(j^3)).

cp's OEIS Frontend

A302354 Expansion of (Sum_{i>=1} x^prime(i))*(Sum_{j>=0} x^(j^3)).

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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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A365290 a(n) is the least positive integer that can be expressed as the sum of a prime number and a positive cube in exactly n ways.

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A307647 Numbers that are the sum of a prime number and a positive cube.

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