OFFSET
0,4
COMMENTS
The smallest (n+1)-digit septuplet is given by 10^n + a(n) + D, with either D = {0, 2, 6, 8, 12, 18, 20} or D = {0, 2, 8, 12, 14, 18, 20}. (For septuplets of the first resp. second type, the first member always ends in digit 1, resp. 9.)
Numerical evidence strongly suggests the conjecture that 0 < a(n) < 10^n for all n > 4, but not even the existence of infinitely many prime septuplets is proved.
Terms up to n = 200 and some further isolated terms due to Norman Luhn et al., cf. LINKS.
LINKS
M. F. Hasler, Table of n, a(n) for n = 0..199
Norman Luhn, Primzahltupel, prime k-tuple: Smallest-n-digit-prime-septuplets, on mathematikalpha.de, Feb 02 2020
EXAMPLE
a(0) = 0 because no single-digit prime starts a prime septuplet.
a(1) = 1 because 10^1 + 1 = 11 = A022009(1) is the first member of the smallest (2-digit) prime septuplet {11, 13, 17, 19, 23, 29, 31} (of the first type).
a(2) = 0 because there is no prime septuplet starting with a 3-digit prime.
a(3) = 4639 because 10^3 + a(3) = 5639 = A022010(1) is the first 4-digit initial member of a prime septuplet, which happens to be of the second type, D = {0, 2, 8, 12, 14, 18, 20}. Similarly, 10^4 + a(4) = 88799 = A022010(2) starts the smallest 5-digit prime septuplet.
For all subsequent terms, a(n) < 10^n (conjectured), so the primes are of the form 10...0XXX where XXX = a(n).
PROG
(PARI) apply( {A343637(n, D=[2, 6, 8, 12, 14, 18, 20], X=2^6+2^14)=forprime(p=10^n, 10^(n+1), my(t=2); foreach(D, d, ispseudoprime(p+d)||(t-- && bittest(X, d))||next(2)); return(p-10^n))}, [0..10]) \\ For illustration; unoptimized code, becomes slow for n >= 11.
CROSSREFS
KEYWORD
nonn,base
AUTHOR
M. F. Hasler, Jul 13 2021
STATUS
approved