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 A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime 1
 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 5, 4, 1, 2, 4, 1, 4, 1, 2, 1, 2, 1, 6, 1, 4, 2, 4, 3, 6, 3, 2, 4, 2, 5, 6, 4, 5, 4, 5, 5, 8, 7, 4, 7, 7, 6, 7, 4, 6, 7, 5, 6, 3, 11, 7, 1, 5, 3, 5, 6, 6, 10, 4, 13, 12, 9, 4, 9, 10, 5, 8, 3, 6, 7, 5, 4, 8, 13, 6, 3, 5, 5, 11, 6, 13, 4, 9, 10, 8, 12, 11, 8, 7, 10, 8, 7, 8, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: a(n)>0 for every n=1,2,3,.... Moreover, any odd integer greater than 2092 can be written as x+y (x,y>0) with x-3, x+3 and x^18+3*y^18 all prime. This has been verified for n up to 2*10^6. Zhi-Wei Sun also made the following general conjecture: For each positive integer m, any sufficiently large odd integer n can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime (and hence there are infinitely many primes in the form x^m+3*y^m). In particular, for m = 1, 2, 3, 4, 5, 6, 18 any odd integer greater than one can be written as x+y (x,y>0) with x^m+3*y^m prime, and for m =1, 2, 3 any odd integer n>15 can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime. Our computation suggests that for each m=7,...,20 any odd integer greater than 32, 10, 24, 30, 48, 36, 72, 146, 48, 48, 152, 2, 238, 84 respectively can be written as x+y (x,y>0) with x^m+3*y^m prime. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..5000 Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588. EXAMPLE a(3)=1 since 2*3-1=5=1+4 with 1^18+3*4^18=206158430209 prime. MATHEMATICA a[n_]:=a[n]=Sum[If[PrimeQ[k^18+3*(2n-1-k)^18]==True, 1, 0], {k, 0, 2n-1}] Do[Print[n, " ", a[n]], {n, 1, 100}] CROSSREFS Cf. A220554, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923. Sequence in context: A286361 A357849 A097295 * A083896 A332083 A084115 Adjacent sequences: A220569 A220570 A220571 * A220573 A220574 A220575 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 16 2012 STATUS approved

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Last modified August 15 11:07 EDT 2024. Contains 375173 sequences. (Running on oeis4.)