cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A365289 a(n) is the least positive integer that can be expressed as the sum of a prime number and a nonnegative cube in exactly n ways.

Original entry on oeis.org

1, 2, 3, 67, 829, 1787, 6654, 8941, 22193, 36277, 57139, 59455, 85377, 158435, 240074, 253628, 313407, 405925, 548802, 891845, 809384, 1317788, 1547004, 2049122, 1838349, 2516848, 3192927, 2448059, 4132313, 4349417, 4438311, 6483753, 6956437, 6175237, 9393491
Offset: 0

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Author

Ilya Gutkovskiy, Aug 31 2023

Keywords

Examples

			For n = 3: 67 = 3 + 4^3 = 59 + 2^3 = 67 + 0^3.

Links

Crossrefs

Cf. A000040, A000578, A302354, A365288, A365290, A365291.

Extensions

More terms from Rémy Sigrist, Sep 07 2023

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Aug 24 2023

Keywords

Crossrefs

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

Programs

  • Mathematica
    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

Formula

G.f.: (Sum_{i>=1} x^prime(i)) * (Sum_{j>=0} x^(j^4)).

A307646 Numbers that are the sum of a prime number and a nonnegative cube.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 29, 30, 31, 32, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 79, 80, 81, 83, 84, 86, 87, 88, 89, 90, 91
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 19 2019

Keywords

Crossrefs

Cf. A000040, A000578, A014089, A045911, A302354, A307647.

Formula

Exponents in expansion of (Sum_{i>=1} x^prime(i)) * (Sum_{j>=0} x^(j^3)).

A356295 Numbers that are not the sum of a nonnegative cube and a prime.

Original entry on oeis.org

1, 9, 16, 22, 26, 28, 33, 35, 36, 52, 57, 63, 65, 76, 78, 82, 85, 92, 96, 99, 112, 118, 119, 120, 122, 126, 129, 133, 141, 146, 155, 160, 169, 170, 183, 185, 188, 202, 209, 210, 216, 217, 225, 236, 244, 246, 248, 267, 273, 280, 286, 300, 302, 309, 326, 328, 330, 342
Offset: 1

Views

Author

Jianing Song, Aug 03 2022

Keywords

Comments

It is conjectured that the subsequence of noncube terms, A045911, is finite (has 6195 terms). But there are infinitely many cubes in this sequence: k^3 if a term if and only if k^3 - (k-1)^3 = 3*k^2 - 3*k + 1 is a nonprime (k-1 is in A257772). For example, for k == 2, 6 (mod 7), 3*k^2 - 3*k + 1 is divisible by 7, so k^3 is a term for k == 2, 6 (mod 7) and k > 2.

Examples

			9 is a term since neither 9 - 0^3 = 9 nor 9 - 1^3 = 8 is a prime.

Crossrefs

Indices of 0 in A302354.
Equals A045911 U {(A257772(n)+1)^3}.
Cf. A014090.

Programs

  • PARI
    isA356295(n) = for(m=0, sqrtnint(n,3), if(isprime(n-m^3), return(0))); return(1)
Showing 1-4 of 4 results.

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