cp's OEIS Frontend

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

%I A350826 #24 Jan 21 2022 14:35:19
%S A350826 1,1,0,0,3,0,13,64,235,1296,7013,41782,253420,1607418,10520883,
%T A350826 70785653,488096844
%N A350826 Number of prime sextuplets with n-digit initial term (A022008).
%C A350826 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.
%C A350826 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.
%H A350826 Norman Luhn, <a href="http://www.pzktupel.de/counting/PI_06.html">PI_6(10^n)</a>
%F A350826 a(n) = # { p in A022008 | 10^(n-1) < p < 10^n }.
%e A350826 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.
%e A350826 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.
%e A350826 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.
%e A350826 For n = 5, {16057, 19417, 43777} are the only 5-digit primes which are initial members of a prime sextuplet, therefore a(5) = 3.
%o A350826 (PARI) apply( {A350826(n,L=10^n)=n=L\10; for(c=0,oo, L<(n=next_A022008(n)) && return(c))}, [1..8])
%Y A350826 Cf. A022008 (initial members of prime sextuplets), A033874 (10^n - precprime(10^n)).
%Y A350826 Cf. A063501, A343636.
%K A350826 nonn,base,more,hard
%O A350826 1,5
%A A350826 _M. F. Hasler_, Jan 17 2022
%E A350826 a(10)-a(12) from _David A. Corneth_, Jan 17 2022
%E A350826 a(13)-a(17) from _Hugo Pfoertner_, Jan 21 2022