A326233 - OEIS

%I #16 Jun 19 2019 18:30:30

%S 4,6,24,29,41,44,74,149,151,216,229,234,240,251,284,415,481,561,574,

%T 704,719,735,751,756,776,819,966,1026,1030,1114,1245,1459,1474,1524,

%U 1535,1584,1749,1936,2035,2084,2101,2165,2189,2241,2246,2251,2301,2305,2384,2511,2541,2710,2865,2955,2990

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

%C 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).)

%C 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.)

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

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

%H Robert Israel, <a href="/A326233/b326233.txt">Table of n, a(n) for n = 1..10000</a>

%H A. Dinculescu, <a href="http://www.utgjiu.ro/math/sma/v13/p13_11.pdf">On the Numbers that Determine the Distribution of Twin Primes</a>, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

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

%p filter:= proc(n) local m;

%p   m:= (7*n)^3;

%p isprime(6*m+1) and isprime(6*m-1)

%p end proc:

%p select(filter, [$1..3000]); # _Robert Israel_, Jun 17 2019

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

%Y 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).

%K nonn

%O 1,1

%A _M. F. Hasler_ and _Antonie Dinculescu_, Jun 14 2019