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 A240697 The smallest p2-p1 when 2n = p1+p4 = p2+p3, where p1 < p2 <= p3 < p4 are prime numbers. 1
 0, 0, 0, 0, 2, 0, 4, 2, 2, 4, 2, 2, 4, 6, 2, 10, 2, 2, 12, 6, 2, 4, 2, 2, 4, 6, 2, 6, 6, 2, 12, 2, 2, 24, 6, 2, 4, 2, 2, 6, 6, 2, 4, 12, 2, 6, 6, 4, 6, 6, 2, 4, 2, 2, 4, 2, 2, 4, 6, 2, 6, 6, 2, 12, 6, 2, 4, 6, 2, 6, 2, 2, 6, 6, 2, 10, 2, 2, 12, 6, 2, 6, 6, 2, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS a(n) is defined as 0 when 2n cannot be written as the sum of two prime numbers in two or more distinct ways. Note that there are only two values of this sequence that are greater than 12 for n<=10000, which are a(34)=24 and a(229)=18. From testing up to n=300000, it appears that only limited terms take a value other than 2, 4, or 6.  Here is a list: a(n) = 0 for n = 1, 2, 3, 4, and 6; a(n) = 10 for n = 16, 76, and 181; a(n) = 12 for n = 19, 31, 44, 64, 79, 86, 124, 139, 259, 269, 274, 314, 391, 404, 484, 499, 604, 619, 691, 709, 859, 1054, 1129, 1399, 1469, 1559, 1574, 2729, 4006, and 5804; a(229) = 18; and a(34) = 24. Conjecture: except for those listed above, no terms in this sequence take a value other than 2, 4, or 6. LINKS Lei Zhou, Table of n, a(n) for n = 1..10000 EXAMPLE When n<5, 2n cannot be written as the sum of two prime numbers in two distinct ways, so a(1) = a(2) = a(3) = a(4) = 0. When n=5, 2n=10 = 3+7 = 5+5, 5-3=2, so a(5)=2. ... When n=18, 2n=36 = 5+31 = 7+29 = 13+23 = 17+19. The smallest difference among the four numbers {5,7,13,17} is 2 = 7-5. So a(18)=2. MATHEMATICA Table[p = 2; While[! PrimeQ[2*n - p], p = NextPrime[p]]; p1 = p; c = 2*n; While[p1 = NextPrime[p1]; (2*p1) <= 2*n, If[PrimeQ[2*n - p1], c = Min[c, p1 - p]; p = p1]]; If[c == 2*n, c = 0]; c, {n, 1, 85}] CROSSREFS Cf. A000040, A002375. Sequence in context: A274706 A037035 A159984 * A271230 A112824 A195133 Adjacent sequences:  A240694 A240695 A240696 * A240698 A240699 A240700 KEYWORD nonn,easy AUTHOR Lei Zhou, Apr 10 2014 STATUS approved

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Last modified July 25 15:18 EDT 2021. Contains 346290 sequences. (Running on oeis4.)