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

1,5

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.

Links

Table of n, a(n) for n=1..17.

Norman Luhn, PI_6(10^n)

Formula

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

Example

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.

Prog

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

Crossrefs

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

Cf. A063501, A343636.

Sequence in context: A186748 A222754 A181905 * A008403 A138349 A369918

Adjacent sequences: A350823 A350824 A350825 * A350827 A350828 A350829

Keyword

nonn,base,more,hard

Author

M. F. Hasler, Jan 17 2022

Extensions

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

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

Status

approved