A152787 -id:A152787 - cp's OEIS frontend

A152788 Integers k such that (k^3)/3 is the average of a pair of twin primes.

6, 30, 84, 144, 186, 204, 270, 360, 516, 576, 726, 756, 810, 990, 1020, 1140, 1446, 1500, 1836, 2010, 2250, 2304, 2820, 3204, 3366, 3564, 4170, 4320, 4344, 4416, 4590, 4656, 5160, 5220, 5820, 5976, 6120, 6204, 6276, 6534, 6876, 7260, 7710, 7806, 7866, 8256

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 12 2008

nonn

These are the integers of the form (3*A014574(i))^(1/3), any index i. - R. J. Mathar, Dec 14 2008

			6 is a term since (6^3)/3 = 72 and (71, 73) are twin primes.
30 is a term since (30^3)/3 = 9000 and (8999, 9001) are twin primes.

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

Cf. A014574, A152786, A152787.

[k:k in [3..9000 by 3]| IsPrime(k^3 div 3 -1) and IsPrime(k^3 div 3 +1)]; // Marius A. Burtea, Jan 01 2020

lst1={}; lst2={}; Do[ p1=Prime[n]; p2=Prime[n+1]; If[p2-p1==2, e=(3*(p1+1))^(1/3); i=Floor[e]; If[e==i, AppendTo[lst1,(p1+1)]; AppendTo[lst2,i]]], {n,2*10!}]; Print[lst1]; Print[lst2]
fQ[n_] := PrimeQ[n^3/3 - 1] && PrimeQ[n^3/3 + 1]; lst = {}; Do[If[fQ@n, AppendTo[lst, n]], {n, 3, 10^4, 3}]; lst

Edited and extended by Robert G. Wilson v, Dec 14 2008

Corrected divisor in definition. - R. J. Mathar, Dec 20 2008

A370453 Twin prime pair sums that equal a twin prime pair product plus 1 (divided by 36).

Original entry on oeis.org

36, 144, 1764, 5184, 360000, 412164, 777924, 4536900, 5673924, 7225344, 12659364, 12830724, 20684304, 37601424, 56972304, 64160100, 81757764, 179506404, 194100624, 255104784, 309689604, 366339600, 461906064, 689062500, 689692644, 1191078144, 1495368900, 1538835984

Offset: 1

Keith F. Lynch, Feb 18 2024

nonn

A twin prime pair (other than {3,5}) is always in the form {6m-1,6m+1}, so the product of the pair is always in the form 36*m^2-1 and a twin prime sum is always in the form 12m. As such, a twin prime sum can be one more than a twin prime product, but not vice versa, nor can a sum and product ever be equal.

{71,73} and {881,883} appear both as sums and as products.

			144 is a term because 71+73 = 144 and 11*13 = 143.
5184 is a term because 2591+2593 = 5184 and 71*73 = 5183.

Keith F. Lynch, Table of n, a(n) for n = 1..197

Subset of A037072.

Cf. A152787.

With[{p = Select[Prime[Range[4200]], PrimeQ[# + 2] &]}, Select[p*(p + 2) + 1, And @@ PrimeQ[#/2 + {-1, 1}] &]] (* Amiram Eldar, Feb 19 2024 *)

cp's OEIS Frontend

A152788 Integers k such that (k^3)/3 is the average of a pair of twin primes.

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