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A303932
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Number of ways to write 2*n as p + 2^k + 3^m, where p is a prime with 11 a quadratic residue modulo p, and k and m are nonnegative integers.
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11
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0, 1, 1, 1, 3, 4, 2, 3, 3, 1, 3, 5, 2, 1, 4, 2, 1, 4, 3, 4, 4, 2, 3, 7, 4, 2, 6, 3, 2, 4, 4, 3, 3, 2, 4, 6, 2, 1, 6, 2, 2, 6, 5, 6, 5, 5, 6, 8, 3, 5, 8, 5, 3, 7, 6, 5, 7, 6, 9, 7, 5, 7, 7, 3, 5, 9, 5, 7, 9, 6, 11, 10, 5, 11, 10, 4, 5, 13, 3, 5
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OFFSET
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1,5
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COMMENTS
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Conjecture 1. a(n) > 0 for all n > 1, i.e., any even number greater than two can be written as the sum of a power 2, a power of 3 and a prime p with 11 a quadratic residue modulo p.
Conjecture 2. For any integer n > 2, we can write 2*n as p + 2^k + 3^m, where p is a prime with 11 a quadratic nonresidue modulo p, and k and m are nonnegative integers.
We have verified Conjectures 1 and 2 for n up to 5*10^8.
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LINKS
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Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)
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EXAMPLE
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a(2) = 1 since 2*2 = 2 + 2^0 + 3^0 with 11 a quadratic residue modulo the prime 2.
a(3) = 1 since 2*3 = 2 + 2^0 + 3^1 with 11 a quadratic residue modulo the prime 2.
a(10) = 1 since 2*10 = 7 + 2^2 + 3^2 with 11 a quadratic residue modulo the prime 7.
a(14) = 1 since 2*14 = 19 + 2^3 + 3^0 with 11 a quadratic residue modulo the prime 19.
a(17) = 1 since 2*17 = 5 + 2^1 + 3^3 with 11 a quadratic residue modulo the prime 5.
a(38) = 1 since 2*38 = 37 + 2^1 + 3^3 with 11 a quadratic residue modulo the prime 37.
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MATHEMATICA
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PQ[n_]:=PQ[n]=n==2||(n>2&&PrimeQ[n]&&JacobiSymbol[11, n]==1);
tab={}; Do[r=0; Do[If[PQ[2n-2^k-3^m], r=r+1], {k, 0, Log[2, 2n-1]}, {m, 0, Log[3, 2n-2^k]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]
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CROSSREFS
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Cf. A000040, A000079, A000244, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660, A303702, A303821.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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