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

1,1

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.

Formula

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

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

Prog

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

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).

Sequence in context: A123046 A237748 A359863 * A087784 A174197 A071224

Adjacent sequences: A326230 A326231 A326232 * A326234 A326235 A326236

Keyword

nonn

Author

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

Status

approved