A336584 Numbers k with Goldbach partitions (p,q) and (r,s) such that k | (p*q -1) and k | (r*s +1).

3770, 4082, 15530, 28730, 38450, 47170, 52490, 59930, 64090, 67730, 79570, 91130, 95290, 95602, 98930, 110890, 111938, 117130, 153842, 168370, 170930, 199810, 204914, 226810, 229970, 236770, 238570, 249290, 250354, 263330, 266330, 268970, 272290, 359210, 359482, 361930

Offset

1,1

Comments

k == +/- 2 (mod 12) == {2,10} (mod 24).

Links

Table of n, a(n) for n=1..36.

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

Formula

Intersection of A336582 & A336583.

Example

3770 is in the sequence since it has a Goldbach partitions, (307,3463) such that 3770 | 307*3463 -1 = 1063140 = 282*3770 and 3770 | (1741*2029 +1) = 3532490 = 937*3770;
4082 is in the sequence since it has a Goldbach partitions, (499,3581) such that 4082 | 499*3583 -1 = 1787916 = 438*4082 and 4082 | (313*3769 +1) = 1179698 = 289*4082; etc.

Mathematica

fQ[n_] := Block[{ flg1 = flg2 = 0, m, p = 3, q}, While[ 2p +1 < n, q = n - p; If[ PrimeQ@q, m = Mod[p*q, n]; If[m == 1, flg1 = 1]; If[m + 1 == n, flg2 = 1]; If[ flg1 == flg2 == 1, Goto[fini]]]; p = NextPrime@p]; Label[fini]; 2p +1 < n]; k = 2; lst = {}; While[k < 400001, If[ fQ@k, AppendTo[lst, k]]; k += 2]; lst

Crossrefs

Cf. A335495, A336582, A336583.

Sequence in context: A338545 A284079 A108179 * A367791 A364141 A364766

Adjacent sequences: A336581 A336582 A336583 * A336585 A336586 A336587

Keyword

nonn

Author

Wesley Ivan Hurt and Robert G. Wilson v, Jul 22 2020

Status

approved