A069496 -id:A069496 - cp's OEIS frontend

A173560 Numbers m such that (6*m)^5 is a sum of a twin prime pair.

16, 44, 84, 135, 161, 631, 849, 880, 1035, 1086, 1721, 1815, 2155, 2704, 2871, 2975, 3011, 3159, 3220, 3365, 3390, 3669, 3996, 4075, 4704, 4761, 5025, 5090, 5299, 5585, 5640, 5970, 6314, 6606, 7035, 7785, 8104, 8129, 8610, 9116, 9665, 9966, 10249

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Feb 21 2010

nonn

The twin prime pairs are characterized in A173255.
No such m has least significant digit (LSD) e = 2 or 7 because a = (6 * e)^5/2 - 1, representing the smaller of the twin primes, would get LSD 5.
No such m has LSD e = 3 or 8, because a+2 = (6 * e)^5/2 + 1, representing the larger prime, would get LSD 5.
The primes in this sequence here are a(6) = 631 = prime(115), a(11) = 1721 = prime(268),
a(17) = 3011 = prime(432), a(49) = 10859 = prime(1320), ...

			p = (6 * 16)^5/2 - 1 = 4076863487 = A000040(193435931); p+2 = A000040(193435932), so a(1) = 16.
p = (6 * 44)^5/2 - 1 = 641194278911 = A000040(24524572848); p+2 = A000040(24524572849), so a(2) = 44.
p = (6 * 84)^5/2 - 1 = 16260080320511 = A000040(553382827197); p+2 = A000040(553382827198), so a(3) = 84.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A001359, A061308, A069496, A172271, A172494, A173255.

Select[Range[700],AllTrue[((6*#)^5-2)/2+{0,2},PrimeQ]&] (* Harvey P. Dale, Dec 21 2024 *)

isok(m) = {my(k = (6*m)^5/2); isprime(k-1) && isprime(k+1);} \\ Amiram Eldar, Jul 19 2025

A226599 Numbers which are the sum of two squared primes in exactly four ways (ignoring order).

Original entry on oeis.org

10370, 10730, 11570, 12410, 13130, 19610, 22490, 25010, 31610, 38090, 38930, 39338, 39962, 40970, 41810, 55250, 55970, 59330, 59930, 69530, 70850, 73730, 76850, 77090, 89570, 98090, 98930, 103298, 118898, 125450, 126290, 130730, 135218, 139490

Offset: 1

Henk Koppelaar, Jun 13 2013

nonn

It appears that all first differences are divisible by 24. - Zak Seidov, Jun 14 2013

			10370 = 13^2 + 101^2 = 31^2 + 97^2 = 59^2 + 83^2 = 71^2 + 73^2.
10730 = 11^2 + 103^2 = 23^2 + 101^2 = 53^2 + 89^2 = 67^2 + 79^2.

Stan Wagon, Mathematica in Action, Springer, 2000 (2nd ed.), Ch. 17.5, pp. 375-378.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A054735 (restricted to twin primes), A037073, A069496.
Cf. A045636 (sum of two squared primes is a superset).
Cf. A214511 (least number having n representations).
Cf. A225104 (numbers having at least three representations is a superset).
Cf. A226539, A226562 (sums decomposed in exactly two and three ways).

Prime2PairsSum := s -> select(x ->`if`(andmap(isprime, x), true, false),
   numtheory:-sum2sqr(s)):
for n from 2 to 10^6 do
  if nops(Prime2PairsSum(n)) = 4 then print(n, Prime2PairsSum(n)) fi;
od;

(* Assuming mod(a(n),24) = 2 *) Reap[ For[ k = 2, k <= 2 + 240000, k = k + 24, pr = Select[ PowersRepresentations[k, 2, 2], PrimeQ[#[[1]]] && PrimeQ[#[[2]]] &]; If[Length[pr] == 4 , Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jun 14 2013 *)

a(n) = p^2 + q^2; p, q are (not necessarily different) primes

A086776 Smaller member of a prime pair (n, n+6) with a square sum.

Original entry on oeis.org

5, 47, 6047, 24197, 31247, 51197, 84047, 151247, 204797, 273797, 387197, 470447, 708047, 806447, 938447, 1804997, 1920797, 1940447, 2060447, 2121797, 2184047, 3150047, 3699197, 6771197, 7411247, 7644047, 8404997, 8652797, 10170047

Offset: 1

Jason Earls, Aug 03 2003

nonn

			6047 is a term because it is the smaller member of the prime pair (6047, 6053) and 6047 + 6053 = 12100 = 110^2.

Cf. A023201, A069496.

Offset 1 from Alois P. Heinz, Jul 27 2019

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A173560 Numbers m such that (6*m)^5 is a sum of a twin prime pair.

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A226599 Numbers which are the sum of two squared primes in exactly four ways (ignoring order).

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A086776 Smaller member of a prime pair (n, n+6) with a square sum.

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