A343637 - OEIS

%I #21 Aug 04 2021 07:40:20

%S 0,1,0,4639,78799,65701,68701,1900501,24066079,12986041,5758291,

%T 63497419,126795511,85452991,693558301,1539113749,1265954431,

%U 959416471,8269773991,620669029,9487038451,1024481911,8285411491,21631441411,15981152869,23307738889,32551582849,114332503171

%N 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.

%C 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.)

%C 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.

%C Terms up to n = 200 and some further isolated terms due to _Norman Luhn_ et al., cf. LINKS.

%H M. F. Hasler, <a href="/A343637/b343637.txt">Table of n, a(n) for n = 0..199</a>

%H Norman Luhn, <a href="https://mathematikalpha.de/primzahltupel">Primzahltupel, prime k-tuple: Smallest-n-digit-prime-septuplets</a>, on mathematikalpha.de, Feb 02 2020

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

%e a(0) = 0 because no single-digit prime starts a prime septuplet.

%e 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).

%e a(2) = 0 because there is no prime septuplet starting with a 3-digit prime.

%e 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.

%e For all subsequent terms, a(n) < 10^n (conjectured), so the primes are of the form 10...0XXX where XXX = a(n).

%o (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.

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

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

%K nonn,base

%O 0,4

%A _M. F. Hasler_, Jul 13 2021