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 A232194 Number of ways to write n = x + y (x, y > 0) with n*x + y and n*y - x both prime. 2
 0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 3, 3, 2, 3, 4, 2, 4, 2, 3, 1, 5, 4, 4, 1, 4, 3, 8, 3, 7, 2, 6, 3, 7, 4, 9, 3, 5, 4, 6, 3, 8, 4, 7, 5, 8, 3, 7, 4, 6, 3, 8, 3, 8, 2, 12, 4, 9, 4, 9, 4, 10, 3, 9, 7, 10, 5, 9, 4, 10, 4, 6, 5, 8, 3, 7, 5, 11, 7, 9, 8, 11, 5, 11, 8, 13, 4, 9, 5, 8, 7, 12, 6, 9, 5, 15, 7, 10, 5, 15, 10 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: (i) a(n) > 0 for all n > 2. Also, a(n) = 1 only for n = 3, 4, 6, 20, 24. (ii) Any positive integer n not among 1, 30, 54 can be written as x + y (x, y > 0) with n*x + y and n*y + x both prime. (iii) Each integer n > 1 not equal to 8 can be expressed as x + y (x, y > 0) with n*x^2 + y (or x^4 + n*y) prime. (iv) Any integer n > 5 can be written as p + q (q > 0) with p and n*q^2 + 1 both prime. See also A232174 for a similar conjecture. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588. EXAMPLE a(3) = 1 since 3 = 1 + 2 with 3*1 + 2 = 3*2 - 1 = 5 prime. a(4) = 1 since 4 = 1 + 3 with 4*1 + 3 = 7 and 4*3 - 1 = 11 both prime. a(6) = 1 since 6 = 1 + 5 with 6*1 + 5 = 11 and 6*5 - 1 = 29 both prime. a(20) = 1 since 20 = 9 + 11 with 20*9 + 11 = 191 and 20*11 - 9 = 211 both prime. a(24) = 1 since 24*19 + 5 = 461 and 24*5 - 19 = 101 both prime. MATHEMATICA a[n_]:=Sum[If[PrimeQ[n*x+(n-x)]&&PrimeQ[n*(n-x)-x], 1, 0], {x, 1, n-1}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000040, A218585, A218654, A219864, A220413, A227898, A227899, A232174, A232186. Sequence in context: A053276 A064065 A284486 * A054705 A025800 A029258 Adjacent sequences:  A232191 A232192 A232193 * A232195 A232196 A232197 KEYWORD nonn AUTHOR Zhi-Wei Sun, Nov 20 2013 STATUS approved

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Last modified July 1 11:07 EDT 2022. Contains 354972 sequences. (Running on oeis4.)