A037073 -id:A037073 - cp's OEIS frontend

A154775 Numbers k such that 13(6k)^2 is the average of a twin prime pair.

2, 4, 5, 42, 46, 49, 59, 82, 84, 100, 119, 128, 137, 182, 185, 187, 192, 233, 264, 301, 303, 340, 376, 390, 395, 422, 438, 446, 471, 472, 494, 518, 527, 570, 598, 609, 611, 633, 667, 688, 714, 716, 726, 728, 733, 744, 831, 837, 865, 875, 896, 926, 940, 948

Offset: 1

M. F. Hasler, Jan 15 2009

nonn

Inspired by Zak Seidov's post to the SeqFan list, cf. link: This yields A154675 as 468 a(n)^2. Indeed, if N/13 is a square, then N=13 k^2 and this can't be the average of a twin prime pair unless k=6m.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zak Seidov, "A154676", Jan 15 2009

Cf. A014574, A037073, A154331, A154772, A154773, A154774, A154675.

okQ[n_]:=Module[{av=468n^2},PrimeQ[av-1]&&PrimeQ[av+1]]; Select[Range[1000],okQ] (* Harvey P. Dale, Jan 21 2011 *)

for(i=1,999, isprime(468*i^2+1) & isprime(468*i^2-1) & print1(i","))

a(n) = sqrt(A154675(n)/468).

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

A363967 Numbers whose divisors can be partitioned into two disjoint sets whose both sums are squares.

Original entry on oeis.org

1, 3, 9, 22, 27, 30, 40, 63, 66, 70, 81, 88, 90, 94, 115, 119, 120, 138, 153, 156, 170, 171, 174, 184, 189, 190, 198, 210, 214, 217, 232, 264, 265, 270, 280, 282, 310, 318, 322, 323, 343, 345, 357, 360, 364, 376, 382, 385, 399, 400, 414, 416, 462, 468, 472, 495, 497

Offset: 1

Amiram Eldar, Jun 30 2023

nonn

If one of the two sets is empty then the term is a number whose sum of divisors is a square (A006532).
If k is a number such that (6*k)^2 is the sum of a twin prime pair (A037073), then (18*k^2)^2 - 1 is a term.
3 is the only prime term.

			9 is a term since its divisors, {1, 3, 9}, can be partitioned into the two disjoint sets, {1, 3} and {9}, whose sums, 1 + 3 = 4 = 2^2 and 9 = 3^2, are both squares.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A000290, A037073.
Subsequence of A333911.
A006532 is a subsequence.
Similar sequences: A333677, A360694.

sqQ[n_] := IntegerQ[Sqrt[n]]; q[n_] := Module[{d = Divisors[n], s, p}, s = Total[d]; p = Position[Rest @ CoefficientList[Product[1 + x^i, {i, d}], x], _?(# > 0 &)] // Flatten; AnyTrue[p, sqQ[#] && sqQ[s - #] &]]; Select[Range[500], q]

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A154775 Numbers k such that 13(6k)^2 is the average 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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A363967 Numbers whose divisors can be partitioned into two disjoint sets whose both sums are squares.

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cp's OEIS Frontend

A154775 Numbers k such that 13*(6*k)^2 is the average of a twin prime pair.

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Author

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PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A363967 Numbers whose divisors can be partitioned into two disjoint sets whose both sums are squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A154775 Numbers k such that 13(6k)^2 is the average of a twin prime pair.