cp's OEIS Frontend

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

%I A357052 #25 Oct 26 2022 18:16:45
%S A357052 4,1,1,87,267,357,33,451,2011,2821,10687,2497,5073,5557,15243,7147,
%T A357052 7357,7197,6627,9157,26317,25833,39207,56067,6667,32937,70561,106533,
%U A357052 597,28503,19167,74551,301711,6747,246871,223353,63057,75183,48513,61323,16107,554287,160141,29821,220711,49441
%N A357052 Distance from 10^n to the next prime triplet.
%C A357052 Equivalently, least k > 0 such that either 10^n + k + {0, 2, 6} or 10^n + k + {0, 4, 6} are primes.
%C A357052 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.
%H A357052 Michael S. Branicky, <a href="/A357052/b357052.txt">Table of n, a(n) for n = 0..216</a>
%H A357052 Norman Luhn, <a href="https://groups.io/g/primenumberstheory/message/139">smallest 7000 digit prime triplet</a>, primenumberstheory mailing list at groups.io, Sep 14 2022
%F A357052 a(n) = min{ k>0 | 10^n + k + [0, 6] contains 3 primes }.
%F A357052 a(n) = min A007529 ∩ [10^n, oo) for n > 0.
%e A357052 (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.
%e A357052 (1087, 1091, 1093) is the smallest 4-digit prime triplet, at distance a(3) = 87 from 10^3.
%e A357052 a(6999) = 1141791245437 is the distance from 10^6999 to the smallest 7000 digit prime triplet, of the form (p, p+2, p+6).
%p A357052 f:= proc(n) local p;
%p A357052    for p from 10^n + 1 by 2 do
%p A357052      if p mod 3 = 1 then if isprime(p) and isprime(p+4) and isprime(p+6) then return p-10^n fi
%p A357052      elif p mod 3 = 2 and isprime(p) and isprime(p+2) and isprime(p+6) then
%p A357052 return p-10^n
%p A357052      fi
%p A357052    od;
%p A357052 end proc:
%p A357052 f(0):= 4:
%p A357052 map(f, [$0..45]); # _Robert Israel_, Sep 15 2022
%p A357052 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
%o A357052 (PARI) A357052(n,q=-9,r=-9)=forprime(p=10^n,,p-r<7 && return(r-10^n);[q,r]=[p,q])
%Y A357052 Cf. A007529 (start of prime triplets), A022004 and A022005 (start of prime triples {0,2,6} resp. {0,4,6}), A343635 (same for quintuplets).
%K A357052 nonn,base
%O A357052 0,1
%A A357052 _M. F. Hasler_, Sep 14 2022