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A326233 Numbers n such that N = (7n)^3 is a twin rank (A002822: 6N +- 1 are twin primes). 6
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified June 19 09:37 EDT 2024. Contains 373501 sequences. (Running on oeis4.)