A326233 Numbers n such that N = (7n)^3 is a twin rank (A002822: 6N +- 1 are twin primes).

4, 6, 24, 29, 41, 44, 74, 149, 151, 216, 229, 234, 240, 251, 284, 415, 481, 561, 574, 704, 719, 735, 751, 756, 776, 819, 966, 1026, 1030, 1114, 1245, 1459, 1474, 1524, 1535, 1584, 1749, 1936, 2035, 2084, 2101, 2165, 2189, 2241, 2246, 2251, 2301, 2305, 2384, 2511, 2541, 2710, 2865, 2955, 2990

Offset: 1

Author

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

Keywords

• nonn

Comments

Dinculescu notes that if m^3 > 1 is a twin rank (i.e., in A002822), then m is always a multiple of 7. (Indeed, 6m^3 + 1 == 0 (mod 7) if m == 1, 2 or 4 (mod 7), and 6m^3 - 1 == 0 (mod 7) for m == 3, 5 or 6 (mod 7).)

He asks whether there are other pairs (a, b) different from (5, 2) and (7, 3) such that all twin ranks m^b > 1 are of the form m = a*n. (Of course (5, 2) and (7, 3) imply that (5, 2k), (7, 3k) and (35, 6k) is also such a pair for any k >= 1.)

This sequence lists these m/7 for (a, b) = (7, 3), see A326234 for the numbers m.

See A326231, A326232 for m^2 and A326235, A326236 for m^6.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Crossrefs

Cf. A002822, A326234 ({1} U 7*{a(n)}), A326231 (analog for n^2), A326232, A326235 (analog for n^6), A326236, A326230 (least twin rank n^k > 1 for given k).

Programs

Maple

filter:= proc(n) local m;
  m:= (7*n)^3;
isprime(6*m+1) and isprime(6*m-1)
end proc:
select(filter, [$1..3000]); # Robert Israel, Jun 17 2019

PARI

select( is(n)=!for(s=1,2,ispseudoprime(6*(7*n)^3+(-1)^s)||return), [1..10^4])

Formula

a(n) = A326234(n+1)/7.