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 A309158 The smallest prime, a(n), larger than prime(n) for which every even difference from 2 to prime(n) - 1 occurs at least once for some pair of primes from prime(n) to a(n) inclusive. 0
 5, 11, 13, 23, 31, 47, 47, 53, 67, 67, 73, 101, 101, 107, 113, 131, 139, 151, 151, 151, 173, 179, 193, 193, 227, 227, 233, 241, 241, 283, 283, 293, 293, 313, 313, 353, 353, 353, 353, 397, 397, 397, 421, 421, 421, 461, 461, 467, 467, 503, 503, 503, 521, 563, 569, 599, 599 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS The "prime differences prime" a(n) is the smallest prime greater than prime(n), n > 1, for which every even difference from 2 to prime(n)-1 occurs for some pair of primes from prime(n) to a(n) inclusive. a(n) is at least prime(n) + (prime(n) - 1) = 2 * prime(n) - 1. If the sequence of prime differences primes is infinite, there are infinitely many pairs of primes for each even difference. If there are only finitely many pairs of primes for some even difference, the sequence ends. Ratios a(n)/prime(n), n = 2 to 15 are 1.67, 2.20, 1.86, 2.09, 2.38, 2.76, 2.47, 2.30, 2.31, 2.16, 1.97, 2.46, 2.35, 2.28. Conjecture: The sequence is infinite. Conjecture: There are finitely many values of n with a(n) = 2 * prime(n) - 1. Conjecture: There are infinitely many values of n with a(n) = a(n-1). Conjecture: For all n, a(n) <= 3 * prime(n). (This is true for n <= 101.) LINKS EXAMPLE For n = 4, prime(4) = 7 and 7 - 1 = 6. Check differences for 7 and 11: 11 - 7 = 4. For 7, 11, and 13: 11 - 7 = 4, 13 - 7 = 6, 13 - 11 = 2, so a(4) = 13. Also prime(6) = 13, 13 - 1 =  12. For 13, 17, 19, 23, 29 and 31, 29 - 17 = 12, 23 - 13 = 10, 31 - 23 = 8, 19 - 13 = 6, 17 - 13 = 4, 19 - 17 = 2, and a(6) = 31. MAPLE for n from 2 to 58 do    a := ithprime(n):    for d from 2 by 2 to a - 1 do       p := ithprime(n);       while not isprime(p + d) do            p := nextprime(p)       od;       if p + d > a then a := p + d fi    od;    print(n, a) od: # Peter Luschny, Jul 17 2019 MATHEMATICA For [n=2, n <= 101, n++,      Clear[d]; d=0;      Clear[a]; a=Prime[n];      While[d < Prime[n]-1,            d=d+2;            Clear[m]; m=n;            While[CompositeQ[d+Prime[m]], m++];            If[d+Prime[m] > a, a=d+Prime[m]]];      Print[{n, Prime[n], a, N[a/Prime[n]]}]      ] CROSSREFS Cf. A000040, A006512, A046132, A046117. Sequence in context: A313992 A190620 A079732 * A161540 A153120 A115782 Adjacent sequences:  A309155 A309156 A309157 * A309159 A309160 A309161 KEYWORD nonn AUTHOR Sally Myers Moite, Jul 14 2019 STATUS approved

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Last modified June 16 12:45 EDT 2021. Contains 345057 sequences. (Running on oeis4.)