A343636 10^n + a(n) is the smallest (n+1)-digit initial member of a prime sextuplet, or a(n) = 0 if no such number exists.

6, 87, 0, 0, 6057, 0, 91257, 526557, 12710877, 2054787, 10234947, 6222567, 33407547, 6589467, 44514957, 587445747, 113139567, 816858057, 210160347, 896654097, 1523730387, 1048338207, 1309977027, 2095519077, 3453564567, 1143089517, 735261237, 723151317, 99411447, 77882127

Offset: 0

Author

M. F. Hasler, Jul 13 2021

Keywords

• nonn
• base
• hard

Comments

The smallest (n+1)-digit sextuplet is given as 10^n + a(n) + {0, 4, 6, 10, 12, 16}.

a(0) = 6 and a(1) = 87, i.e., n = 0 and n = 1, are the only cases where larger members of the sextuplet have one digit more than the smallest member of the sextuplet. These terms would be zero if all members of the sextuplet ought to have the same number of digits. We require only the first member to have the given number of digits, in order to include these two nontrivial terms.

Numerical evidence strongly suggests the conjecture that 0 < a(n) < 10^n for all n > 5, but not even the existence of infinitely many prime sextuplets is proved.

Values for n > 300 found by Norman Luhn: a(399) = 33756090918084087, a(499) = 464261549124325347 (October 2020), a(599) = 314360191056418137 (June 2021).

Examples

			a(0) = 6 because A022008(1) = 7 = 10^0 + 6 is the start of the smallest prime sextuplet, starting with a 1-digit prime.
a(1) = 87 because A022008(2) = 97 = 10^1 + 87 is the start of the smallest prime sextuplet starting with a 2-digit prime.
a(n) = 0 iff n = 2, 3 or 5, because there is no prime sextuplet with members having 3, 4 or 6 digits.
a(4) = 6057 because A022008(3) = 16057 = 10^4 + 6057 is the start of the smallest prime sextuplet made of 5-digit primes.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 0..299
• Norman Luhn, Primzahltupel, prime k-tuple: Smallest-n-digit-prime-sexuplets, on mathematikalpha.de, 2020

Crossrefs

Cf. A022008 (start of prime sextuplets).

Programs

PARI

apply( {A343636(n,D=[16,12,10,6,4])=forprime(p=10^n, 10^(n+1), foreach(D, d, ispseudoprime(p+d)||next(2));return(p-10^n))}, [0..11]) \\ For illustration; unoptimized code, very fast only for n < 12.