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 1-6 of 6 results.

A211167 Neither a cube nor the sum of a prime and a positive cube.

Original entry on oeis.org

2, 5, 7, 9, 16, 17, 22, 23, 26, 28, 33, 35, 36, 41, 43, 47, 52, 53, 57, 59, 63, 65, 73, 76, 78, 82, 85, 89, 92, 96, 99, 103, 112, 113, 118, 119, 120, 122, 126, 129, 133, 141, 146, 149, 151, 155, 160, 163, 169, 170, 179, 183, 185, 188, 193, 197
Offset: 1

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Author

Charles R Greathouse IV, Feb 19 2013

Keywords

Comments

Hardy & Littlewood's conjecture that this sequence is finite (Conjecture L, p. 51).
Up to 4*10^9 there are 7050 such numbers, the last one being 78526384. - Giovanni Resta, Feb 20 2013

Links

Crossrefs

Cf. A045911.

Programs

  • PARI
    is(n)=if(ispower(n,3),return(0));for(k=1,n^(1/3),if(isprime(n-k^3), return(0)));1

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

Original entry on oeis.org

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

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Author

Ilya Gutkovskiy, Apr 06 2018

Keywords

Comments

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

Examples

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

Crossrefs

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

Programs

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

Formula

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

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

Views

Author

Ilya Gutkovskiy, Aug 24 2023

Keywords

Crossrefs

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

Programs

  • Mathematica
    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

A236767 Numbers whose square is a fourth power plus a prime.

Original entry on oeis.org

2, 10, 37, 82, 442, 577, 730, 901, 1090, 1297, 1765, 2026, 4357, 5185, 5626, 7570, 8650, 9217, 9802, 10405, 11026, 15130, 17425, 18226, 23410, 24337, 26245, 31330, 34597, 35722, 40402, 41617, 47962
Offset: 1

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Author

Hans Havermann, Jan 30 2014

Keywords

Comments

Based on a 1999 observation of Alessandro Zaccagnini (via John Robertson) intended to dissuade expectation of a finite fourth-power analogy to A020495, A045911.
It can be shown that A089001^2 + 1 are members of this sequence. David Applegate shows that they are the only members: If x^2 = y^4 + p, let a = x - y^2. Then y^4 + p = x^2 = (y^2 + a)^2 = y^4 + 2a*y^2 + a^2, so p = 2a*y^2 + a^2, and so a divides p. Since p is a prime, a must be a unit (that is, +1 or -1). But since p >= 2, a must be +1.

Examples

			2 is a term because 2^2 = 1^4 + 3;
10 is a term because 10^2 = 3^4 + 19;
37 is a term because 37^2 = 6^4 + 73.

Links

Crossrefs

Cf. A020495, A045911, A089001.

Programs

  • Mathematica
    r=Range[10000]^4; j=1; Do[c=i^2; k=c^2-Take[r,i]; Do[c++; j=j+2; k=k+j; If[MemberQ[PrimeQ[k], True], Print[c]], {2*i+1}], {i, 10000}] (* brute force *)
s=A089001; s^2+1 (* based on formula *)

Formula

A089001^2 + 1

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 1-6 of 6 results.

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