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 A343637 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. 5
 0, 1, 0, 4639, 78799, 65701, 68701, 1900501, 24066079, 12986041, 5758291, 63497419, 126795511, 85452991, 693558301, 1539113749, 1265954431, 959416471, 8269773991, 620669029, 9487038451, 1024481911, 8285411491, 21631441411, 15981152869, 23307738889, 32551582849, 114332503171 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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.) 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. Terms up to n = 200 and some further isolated terms due to Norman Luhn et al., cf. LINKS. LINKS M. F. Hasler, Table of n, a(n) for n = 0..199 Norman Luhn, Primzahltupel, prime k-tuple: Smallest-n-digit-prime-septuplets, on mathematikalpha.de, Feb 02 2020 FORMULA a(n) = min { p > 10^n; p in A022009 U A022010 } - 10^n, for n > 2. EXAMPLE a(0) = 0 because no single-digit prime starts a prime septuplet. 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). a(2) = 0 because there is no prime septuplet starting with a 3-digit prime. 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. For all subsequent terms, a(n) < 10^n (conjectured), so the primes are of the form 10...0XXX where XXX = a(n). PROG (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. CROSSREFS Cf. A022009 and A022010 (initial members of prime septuplets of first and second type). Cf. A343635, A343636 (analog for quintuplets and sextuplets). Sequence in context: A139410 A031566 A236815 * A233855 A068268 A227482 Adjacent sequences: A343634 A343635 A343636 * A343638 A343639 A343640 KEYWORD nonn,base AUTHOR M. F. Hasler, Jul 13 2021 STATUS approved

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Last modified September 8 05:26 EDT 2024. Contains 375751 sequences. (Running on oeis4.)