A343637 10^n + a(n) is the least (n+1)-digit prime member of a prime septuplet, or a(n) = 0 if no such number exists.

0, 1, 0, 4639, 78799, 65701, 68701, 1900501, 24066079, 12986041, 5758291, 63497419, 126795511, 85452991, 693558301, 1539113749, 1265954431, 959416471, 8269773991, 620669029, 9487038451, 1024481911, 8285411491, 21631441411, 15981152869, 23307738889, 32551582849, 114332503171

Offset: 0

Author

M. F. Hasler, Jul 13 2021

Keywords

• nonn
• base

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.

Examples

			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).
		

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

Crossrefs

Cf. A022009 and A022010 (initial members of prime septuplets of first and second type).

Cf. A343635, A343636 (analog for quintuplets and sextuplets).

Programs

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.

Formula

a(n) = min { p > 10^n; p in A022009 U A022010 } - 10^n, for n > 2.