A350828 Number of prime octuplets with initial member (A065706) between 10^(n-1) and 10^n.

0, 2, 0, 1, 1, 3, 3, 9, 28, 136, 541, 2936

Offset: 1

Author

M. F. Hasler, Mar 01 2022

Keywords

• nonn
• base
• hard
• more

Comments

"Between 10^(n-1) and 10^n" is equivalent to saying "with n (decimal) digits".

A prime octuplet is a sequence of 8 consecutive primes (p1, ..., p8) of minimal diameter p8 - p1 = 26.

Terms a(1)-a(12) computed from b-file a(1..18123) for A065706. Using Luhn's database, cf. LINKS, one can get 3 more terms.

So far, the last term of all the octuplets has the same number of digits as the initial term.

Examples

			a(1) = a(3) = 0 because there is no single-digit nor a 3-digit prime initial member of a prime octuplet.
a(2) = 2 because 11 and 17 are the only 2-digit members of A065706, i.e., primes to start a prime octuplet.
a(4) = a(5) = 1 because 1277 (resp. 88793) is the only prime with 4 (resp. 5) digits to start a prime octuplet.
Then there are a(6) = 3 six-digit primes, 113147, 284723 and 855713, which start a prime octuplet.
		

Links

• Norman Luhn, The big database of "The smallest prime k-tuplets", section "prime 8-tuplets": complete up to 10^16 as of March 2022.

Crossrefs

Cf. A065706 (initial members p of prime octuplets (p, ..., p+26)), A022011, A022012, A022013 (idem, specifically for each of the three possible patterns).

Cf. A350825, A350826, A350827: similar for quintuplets, sextuplets and septuplets.

Programs

PARI

(D(v)=v[^1]-v[^-1])( [setsearch(A065706,10^n,1) | n<-[0..12]] ) \\ where A065706 is a vector of at least 3660 terms of that sequence.