

A335495


Numbers k with a Goldbach partition (p,q) such that k  (p*q + 1).


4



5, 8, 10, 12, 24, 30, 36, 40, 42, 48, 50, 56, 58, 60, 66, 70, 72, 74, 84, 90, 96, 106, 112, 120, 130, 132, 144, 156, 168, 170, 180, 184, 198, 204, 210, 216, 220, 222, 224, 228, 232, 234, 240, 246, 252, 260, 264, 276, 280, 288, 294, 296, 300, 304, 312, 318, 330, 336, 340
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OFFSET

1,1


COMMENTS

From Robert G. Wilson v, Jul 22 2020: (Start)
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=n2. p*q=2n4 and 2n4 (mod n) == 4. This will only work with 5 since 4 (mod 5) is 1.
Few terms are twice a prime: 10, 58, 74, 106, 562, 1546, 2474, 2554, 2578, 3394, 3418, 3754, 4282, 6242, 6602, 8578, 10306, ..., .
Number of terms less than or equal to 10^n: 3, 21, 149, 1181, 9919, ..., . (End)


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions


EXAMPLE

5 is in the sequence since it has a Goldbach partition, (3,2) such that 5  (3*2  1) = 5.
8 is in the sequence since it has a Goldbach partition, (5,3) such that 8  (5*3 + 1) = 16.
10 is in the sequence since it has a Goldbach partition, (7,3) such that 10  (7*3  1) = 20.
12 is in the sequence since it has a Goldbach partition, (7,5) such that 12  (7*5 + 1) = 36.


MATHEMATICA

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
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 *)


CROSSREFS

Sequence in context: A314381 A325180 A087280 * A280537 A022413 A078781
Adjacent sequences: A335492 A335493 A335494 * A335496 A335497 A335498


KEYWORD

nonn


AUTHOR

Wesley Ivan Hurt, Jun 11 2020


STATUS

approved



