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 1-3 of 3 results.

A130555 Numbers that are sums of sixth powers of two distinct primes.

Original entry on oeis.org

793, 15689, 16354, 117713, 118378, 133274, 1771625, 1772290, 1787186, 1889210, 4826873, 4827538, 4842434, 4944458, 6598370, 24137633, 24138298, 24153194, 24255218, 25909130, 28964378, 47045945, 47046610, 47061506, 47163530
Offset: 1

Views

Author

Jonathan Vos Post, Aug 09 2007

Keywords

Comments

This is to 6th powers as A130292 is to fifth powers, A130873 is to 4th powers and A120398 is to cubes. These can never be prime, as sixth powers are cubes and the sum of cubes factorizations applies. There are semiprimes for values beginning a(1) = 793, a(2) = 15689 = 29 * 541, a(4) = 117713 = 53 * 2221, a(11) = 4826873 = 173 * 27901.

Examples

			a(1) = prime(1)^6 + prime(2)^6 = 2^6 + 3^6 = 64 + 729 = 793 = 13 * 61.
		

Crossrefs

Programs

  • Mathematica
    Select[Sort[Flatten[Table[Prime[n]^6 + Prime[k]^6, {n, 15}, {k, n - 1}]]], # <= Prime[15^6] &]
    Union[Total/@Subsets[Prime[Range[20]]^6,{2}]] (* Harvey P. Dale, Mar 11 2012 *)

Formula

{A001014(A000040(i)) + A001014(A000040(j)) for i > j}.

A132214 Numbers that are sums of seventh powers of two distinct primes.

Original entry on oeis.org

2315, 78253, 80312, 823671, 825730, 901668, 19487299, 19489358, 19565296, 20310714, 62748645, 62750704, 62826642, 63572060, 82235688, 410338801, 410340860, 410416798, 411162216, 429825844, 473087190, 893871867, 893873926
Offset: 1

Views

Author

Jonathan Vos Post, Aug 13 2007

Keywords

Comments

This is to 7th powers as A130555 is to 6th powers, A130292 is to fifth powers, A130873 is to 4th powers and A120398 is to cubes. These can never be prime, as the polynomial x^7 + y^7 factors over Z. Note however that A132215, which is the analog for eighth powers, can be prime, as seen also in A132216.

Examples

			a(1) = 2^7 + 3^7 = 2315 = 5 * 463.
		

Crossrefs

Programs

  • Maple
    P:= select(isprime, [2,seq(i,i=3..100,2)]): nP:= nops(P):
    N:= 2^7 + P[-1]^7:
    sort(convert(select(`<=`, {seq(seq(P[i]^7+P[j]^7,j=i+1..nP),i=1..nP-1)},N),list)); # Robert Israel, Jul 01 2024
  • Mathematica
    Select[Sort[ Flatten[Table[Prime[n]^7 + Prime[k]^7, {n, 15}, {k, n - 1}]]], # <= Prime[15^7] &]
    Union[Total/@(Subsets[Prime[Range[10]],{2}]^7)] (* Harvey P. Dale, Jan 03 2012 *)

Formula

{A001015(A000040(i)) + A001015(A000040(j)) for i > j}.

A132215 Numbers that are sums of eighth powers of two distinct primes.

Original entry on oeis.org

6817, 390881, 397186, 5765057, 5771362, 6155426, 214359137, 214365442, 214749506, 220123682, 815730977, 815737282, 816121346, 821495522, 1030089602, 6975757697, 6975764002, 6976148066, 6981522242, 7190116322, 7791488162
Offset: 1

Views

Author

Jonathan Vos Post, Aug 13 2007

Keywords

Comments

This is to 8th powers as A132214 is to 7th powers, A130555 is to 6th powers, A130292 is to fifth powers, A130873 is to 4th powers and A120398 is to cubes. These CAN be prime, as the polynomial x^8 + y^8 is irreducible over Z, as seen in A132216. The first such example is a(11) = A132216(1) = 2^8 + 13^8 = 256 + 815730721 = 815730977, which is prime.
A subset of A003380. - R. J. Mathar, May 11 2008

Examples

			a(1) = 2^8 + 3^8 = 256 + 6561 = 6817 = 17 * 401.
		

Crossrefs

Programs

  • Mathematica
    Select[Sort[ Flatten[Table[Prime[n]^8 + Prime[k]^8, {n, 15}, {k, n - 1}]]], # <= Prime[15^8] &]
    Total/@Subsets[Prime[Range[10]]^8,{2}]//Sort (* Harvey P. Dale, Jun 27 2017 *)

Formula

{A001016(A000040(i)) + A001016(A000040(j)) for i > j}.
Showing 1-3 of 3 results.