|
| |
|
|
A335495
|
|
Numbers k with a Goldbach partition (p,q) such that k | (p*q +- 1).
|
|
6
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
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.
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
|
|
|
|
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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
