A350830 Number of prime 10-tuples (or decaplets) with initial member (A257127) between 10^(n-1) and 10^n.

0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 4, 15, 81, 357, 1685, 8256

Offset

1,11

Comments

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

A prime 10-tuple (or decaplet) is a sequence of 9 consecutive primes (p1, ..., p10) of minimum possible diameter p10 - p1 = 32.

Terms a(1)-a(16) computed from b-files a(1..10^4) for A027569 and A027570. Using Luhn's data (cf. LINKS) one can obtain a(18) and a(19).

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

Links

Table of n, a(n) for n=1..16.

Norman Luhn, The big database of "The smallest prime k-tuplets", section "10-uplets": up to 10^20 as of March 2022.

Example

a(2) = 1 because 11 is the only two-digit prime to start a prime decaplet, i.e., member of A257127.
a(n) = 0 for all other n < 10 because the next larger prime decaplet is made of 10-digit primes, A257127(2) = 9853497737 and successors.
a(10) = 1 because there is only one prime decaplet made of 10-digit primes.
a(11) = 4 because there are only four terms in A257127 (for indices n = 3..6) which have 11 digits.

Prog

(PARI) (D(v)=v[^1]-v[^-1])( [setsearch(A257127, 10^n, 1) | n<-[0..16]] ) \\ where A257127 is a vector of at least 10400 terms of that sequence.

Crossrefs

Cf. A257127 (initial members p of prime 9-tuples (p, ..., p+32)), A027569, A027570 (idem, specifically for each of the two possible patterns).

Cf. A350825 - A350829: similar for quintuples, sextuples, septuples, octuples and 9-tuples.

Sequence in context: A125307 A073479 A147690 * A068313 A174661 A207161

Adjacent sequences: A350827 A350828 A350829 * A350831 A350832 A350833

Keyword

nonn,base,hard,more

Author

M. F. Hasler, Mar 01 2022

Status

approved