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A343635 10^n + a(n) is the least (n+1)-digit prime member of a prime 5-tuple, or a(n) = 0 if no such number exists.

%I A343635 #39 Mar 19 2023 08:21:18
%S A343635 4,1,1,481,5727,1107,8851,18027,5457,408807,57931,358531,274587,
%T A343635 256497,6111627,67437,3246567,1638811,8224977,11687221,24556351,
%U A343635 3129657,15602131,571381,23034391,110598987,26716321,31722117,39711931,5046777,81054327,1346611,44656587
%N A343635 10^n + a(n) is the least (n+1)-digit prime member of a prime 5-tuple, or a(n) = 0 if no such number exists.
%C A343635 The smallest (n+1)-digit prime 5-tuple is given by 10^n + a(n) + D, with either D = {0, 2, 6, 8, 12} or D = {0, 4, 6, 10, 12}. N = 0 is the only case where the last member of the 5-tuple has one digit more than the first member.
%C A343635 Numerical evidence strongly suggests the conjecture that 0 < a(n) < 10^n for all n > 0, but not even the existence of infinitely many prime 5-tuples is proved.
%C A343635 Some further isolated terms, due to _Norman Luhn_ et al., giving the start of the smallest 500, 600, 700, ..., 1200 digit quintuplets of first or second type:
%C A343635   a(499) = min(58195471283341, 69672492141807),
%C A343635   a(599) = min(319491304676641, 12754947401547),
%C A343635   a(699) = min(2254633393747621, 209264286017367),
%C A343635   a(799) = min(2117758391972791, 1299258655252617),
%C A343635   a(899) = min(2365663735968811, 1484244113736867),
%C A343635   a(999) = min(3554007760224751, 3818999670116007),
%C A343635   a(1099) = min(26317044823878361, 15720821612555937),
%C A343635   a(1199) = min(20483870459152351, 7033048489975137).
%C A343635 Terms through a(399) may be determined by taking the minima of those in the linked tables for quintuplets by _Norman Luhn_ et al. - _Michael S. Branicky_, Jul 24 2021
%C A343635 The first member of the quintuplets of the first type always ends in digit 1 (except for the 5-tuple (5, 7, 11, 13, 17) corresponding to a(0)), for the second type it always ends in digit 7. Therefore all a(n), n > 0, end in a digit 1 or 7, which indicates the type of the 5-tuple, i.e., the set D that has to be added to 10^n + a(n) to get the whole 5-tuple. - _M. F. Hasler_, Aug 04 2021
%H A343635 M. F. Hasler, <a href="/A343635/b343635.txt">Table of n, a(n) for n = 0..399</a> (terms up to a(51) from Michael S. Branicky), Aug 04 2021.
%H A343635 Norman Luhn, <a href="https://mathematikalpha.de/primzahltupel">Primzahltupel, prime k-tuple: Smallest-n-digit-prime-quintuplets</a>, on mathematikalpha.de, 2020. <a href="https://mathematikalpha.de/wp-content/uploads/2020/07/smallest-n-digit-prime-quintuplets.pdf">Tables</a>.
%e A343635 a(0) = 4 because {5, 7, 11, 13, 17} is the smallest prime 5-tuple and it starts with the single-digit prime 10^0 + a(0) = 5 = A022006(1).
%e A343635 a(1) = 1 because 10^1 + 1 = 11 = A022006(2) is the 2-digit prime to start a prime 5-tuple {11, 13, 17, 19, 23}, again of the first type.
%e A343635 a(2) = 1 and a(3) = 481 because 10^2 + 1 = 101 = A022006(3) and 10^3 + 481 = 1481 = A022006(4) are the smallest 3-digit, resp. 4-digit, initial members of a prime 5-tuple, both again of the first type.
%e A343635 a(4) = 5727 because 10^4 + 5727 = 15727 = A022007(6) is the smallest 5-digit initial member of a prime 5-tuple, now of the second type.
%e A343635 It appears that for all n > 0, a(n) < 10^n, so that the primes are of the form 10...0XXX where XXX = a(n) and 0...0 stands for a string of zero or more digits 0.
%o A343635 (PARI) apply( {A343635(n,q=[1..4],i=0)=forprime(p=10^n,, (q[1+i]+12==q[i++]=p) && return(p-12-10^n); i>3 && i=0)}, [0..15]) \\ Shorter but slightly slower (?)
%o A343635 (PARI) apply( {A343635(n, i=ispseudoprime, q)=forprime(p=10^n,, i(p+12) && i(p+6) && (p+6 > q=nextprime(p+2)) && i(q+6) && return(p-10^n))}, [0..15])
%o A343635 (Python)
%o A343635 from sympy import nextprime
%o A343635 def a(n):
%o A343635     p = nextprime(10**n)
%o A343635     q = nextprime(p); r = nextprime(q); s = nextprime(r); t = nextprime(s)
%o A343635     while p < 10**(n+1):
%o A343635         if t - p == 12: return p - 10**n
%o A343635         p, q, r, s, t = q, r, s, t, nextprime(t)
%o A343635     return 0
%o A343635 print([a(n) for n in range(14)]) # _Michael S. Branicky_, Jul 24 2021
%Y A343635 Cf. A022006 and A022007 (initial members of prime 5-tuples of first and second type).
%Y A343635 Cf. A343636, A343637 (analog for sextuplets and septuplets).
%K A343635 nonn,base,hard
%O A343635 0,1
%A A343635 _M. F. Hasler_, Jul 17 2021