A357052 Distance from 10^n to the next prime triplet.

4, 1, 1, 87, 267, 357, 33, 451, 2011, 2821, 10687, 2497, 5073, 5557, 15243, 7147, 7357, 7197, 6627, 9157, 26317, 25833, 39207, 56067, 6667, 32937, 70561, 106533, 597, 28503, 19167, 74551, 301711, 6747, 246871, 223353, 63057, 75183, 48513, 61323, 16107, 554287, 160141, 29821, 220711, 49441

Offset: 0

Author

M. F. Hasler, Sep 14 2022

Keywords

• nonn
• base

Comments

Equivalently, least k > 0 such that either 10^n + k + {0, 2, 6} or 10^n + k + {0, 4, 6} are primes.

The initial term, index n = 0, is the only even term and the only case where the last member of the triplet has one digit more than the first member. The value a(0) = 4 correspond to the prime triplet (5, 7, 11). We do not consider the triplets (2, 3, 5) or (3, 5, 7) which come earlier but do not follow the standard pattern.

Examples

			(11, 13, 17) and (101, 103, 107) are the smallest 2-digit and 3-digit prime triplets, at distance a(1) = a(2) = 1 from 10^1 and 10^2, respectively.
(1087, 1091, 1093) is the smallest 4-digit prime triplet, at distance a(3) = 87 from 10^3.
a(6999) = 1141791245437 is the distance from 10^6999 to the smallest 7000 digit prime triplet, of the form (p, p+2, p+6).
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 0..216
• Norman Luhn, smallest 7000 digit prime triplet, primenumberstheory mailing list at groups.io, Sep 14 2022

Crossrefs

Cf. A007529 (start of prime triplets), A022004 and A022005 (start of prime triples {0,2,6} resp. {0,4,6}), A343635 (same for quintuplets).

Programs

Maple

f:= proc(n) local p;
   for p from 10^n + 1 by 2 do
     if p mod 3 = 1 then if isprime(p) and isprime(p+4) and isprime(p+6) then return p-10^n fi
     elif p mod 3 = 2 and isprime(p) and isprime(p+2) and isprime(p+6) then
return p-10^n
     fi
   od;
end proc:
f(0):= 4:
map(f, [$0..45]); # Robert Israel, Sep 15 2022
A357052 := proc(n) local p,q,r; p,q,r := 10^n,0,0; while p-r <> 6 do p,q,r := nextprime(p+1),p,q; od; r-10^n; end; # M. F. Hasler, Sep 15 2022

PARI

A357052(n,q=-9,r=-9)=forprime(p=10^n,,p-r<7 && return(r-10^n);[q,r]=[p,q])

Formula

a(n) = min{ k>0 | 10^n + k + [0, 6] contains 3 primes }.

a(n) = min A007529 ∩ [10^n, oo) for n > 0.