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%I #14 Aug 15 2020 13:11:18
%S 5,8,10,12,24,30,36,40,42,48,50,56,58,60,66,70,72,74,84,90,96,106,112,
%T 120,130,132,144,156,168,170,180,184,198,204,210,216,220,222,224,228,
%U 232,234,240,246,252,260,264,276,280,288,294,296,300,304,312,318,330,336,340
%N Numbers k with a Goldbach partition (p,q) such that k | (p*q +- 1).
%C From _Robert G. Wilson v_, Jul 22 2020: (Start)
%C 5 is the only odd member. To qualify as a Goldbach partition, an odd number candidate must have as its two primes, p&q, p=2 and q=n-2. p*q=2n-4 and 2n-4 (mod n) == -4. This will only work with 5 since -4 (mod 5) is 1.
%C Few terms are twice a prime: 10, 58, 74, 106, 562, 1546, 2474, 2554, 2578, 3394, 3418, 3754, 4282, 6242, 6602, 8578, 10306, ..., .
%C Number of terms less than or equal to 10^n: 3, 21, 149, 1181, 9919, ..., . (End)
%H Robert G. Wilson v, <a href="/A335495/b335495.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%e 5 is in the sequence since it has a Goldbach partition, (3,2) such that 5 | (3*2 - 1) = 5.
%e 8 is in the sequence since it has a Goldbach partition, (5,3) such that 8 | (5*3 + 1) = 16.
%e 10 is in the sequence since it has a Goldbach partition, (7,3) such that 10 | (7*3 - 1) = 20.
%e 12 is in the sequence since it has a Goldbach partition, (7,5) such that 12 | (7*5 + 1) = 36.
%t Table[If[Sum[Sign[(1 - Ceiling[(i (n - i) + 1)/n] + Floor[(i (n - i) + 1)/n]) + (1 - Ceiling[(i (n - i) - 1)/n] + Floor[(i (n - i) - 1)/n])] (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[n/2]}] > 0, n, {}], {n, 400}] // Flatten
%t fQ = Compile[{{n, _Integer}}, Block[{p = 3, q}, While[q = n - p; m = Mod[p*q, n]; p < q && ! PrimeQ@q || m != 1 && m + 1 != n, p = NextPrime@p]; p < q]]; Join[{5}, Select[ 2Range@ 175, fQ]] (* _Robert G. Wilson v_, Jul 22 2020 *)
%K nonn
%O 1,1
%A _Wesley Ivan Hurt_, Jun 11 2020
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