A350827 -id:A350827 - cp's OEIS frontend

A350825 Number of prime 5-tuples with initial member (A086140) between 10^(n-1) and 10^n.

2, 2, 1, 4, 12, 44, 256, 1062, 5838

Offset: 1

M. F. Hasler, Mar 01 2022

"Between 10^(n-1) and 10^n" is equivalent to saying "with n digits".
For n = 1 and n = 2, the last term of the last 5-tuple in that range (cf EXAMPLE) has one digit more than the initial term.
Terms a(1)-a(9) computed from b-files a(1..10000) for A022006 and A022007.

			a(1) = 2 because there are just two single-digit primes to start a prime 5-tuple, namely 5 = A022006(1) and 7 = A022007(1).
a(2) = 2 because 11 = A022006(2) and 97 = A022007(2) are the only two two-digit primes to start a prime 5-tuple.
a(3) = 1 because there is only one three-digit prime to start a prime 5-tuple, namely 101 = A022006(3).
Then there are a(4) = 4 four-digit primes, 1481, 1867, 3457 and 5647, which start a prime 5-tuple.

Cf. A086140 (initial members p of prime quintuplets), A022006, A022007 (idem, specifically for patterns (p, p+2, ...) resp. (p, p+4, ...)).

Cf. A350826, A350827, A350828: similar for sextuplets, septuplets and octuplets.

(D(v)=v[^1]-v[^-1])( [setsearch(A086140, 10^n, 1) | n<-[0..9]] ) \\ where A086140 is a vector of at least 7221 terms of that sequence.

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

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Mar 01 2022

"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.

			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.

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

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.

(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.

cp's OEIS Frontend

A350825 Number of prime 5-tuples with initial member (A086140) between 10^(n-1) and 10^n.

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