A350826 Number of prime sextuplets with n-digit initial term (A022008).

1, 1, 0, 0, 3, 0, 13, 64, 235, 1296, 7013, 41782, 253420, 1607418, 10520883, 70785653, 488096844

Offset: 1

Author

M. F. Hasler, Jan 17 2022

Keywords

• nonn
• base
• more
• hard

Comments

Prime sextuplets are of the form (p, p+4, p+6, p+10, p+12, p+16), where p is the initial member, listed in A022008.

For n = 1 and n = 2 (see Example), the last member of the sextuplet has one digit more than the initial member (so the count would be 0 for these two, if all terms of the sextuplet had to have the same length). As far as we know, for all n > 2, all members of the sextuplets have the same length. A sufficient condition for this is that A033874(n) > 16.

Examples

			For n = 1, p = 7 is the only 1-digit prime to be the initial term of a prime sextuplet, (7, 11, 13, 17, 19, 23), hence a(1) = 1.
For n = 2, p = 97 is the only 2-digit prime to be the initial term of a prime sextuplet, (97, 101, 103, 107, 109, 113), whence a(2) = 1.
For n = 3 and n = 4, there is no n-digit prime to be the initial term of a prime sextuplet, so a(n) = 0.
For n = 5, {16057, 19417, 43777} are the only 5-digit primes which are initial members of a prime sextuplet, therefore a(5) = 3.
		

Links

• Norman Luhn, PI_6(10^n)

Crossrefs

Cf. A022008 (initial members of prime sextuplets), A033874 (10^n - precprime(10^n)).

Cf. A063501, A343636.

Programs

PARI

apply( {A350826(n,L=10^n)=n=L\10; for(c=0,oo, L<(n=next_A022008(n)) && return(c))}, [1..8])

Formula

a(n) = # { p in A022008 | 10^(n-1) < p < 10^n }.

Extensions

a(10)-a(12) from David A. Corneth, Jan 17 2022

a(13)-a(17) from Hugo Pfoertner, Jan 21 2022