A309780 Even numbers m having the property that for every odd prime divisor p of m there exists a positive integer k < p-1, such that p-k|m-k.

20, 28, 44, 50, 52, 68, 76, 80, 88, 92, 104, 110, 112, 116, 124, 136, 148, 152, 164, 170, 172, 176, 184, 188, 196, 200, 208, 212, 230, 232, 236, 238, 242, 244, 248, 268, 272, 284, 286, 290, 292, 296, 304, 316, 320, 322, 328, 332, 338, 344, 356, 364, 368, 374

Offset: 1

Author

David James Sycamore, Aug 17 2019

Keywords

• nonn

Comments

Subsequence of A309769. Even number m is a term if and only if for every odd prime divisor p, m can be written as 2*r*p, where r >= 2, and p is greater than the smallest prime divisor of 2*r-1.

From above, 4^k*p is a term for every prime p >= 5 and k >= 1. - David A. Corneth, Aug 17 2019

More general than the above, David James Sycamore finds (2*r)^k * p is a term for all r>=2, k>=1 and prime p > q, the smallest prime divisor of 2*r-1. - David A. Corneth, Aug 26 2019

Examples

			20 = 4*5 is a term (k=2 for p=5).
110 = 10*11 = 22*5 is a term (k=8 for p=11 and k=2 for p=5).
		

Crossrefs

Cf. A309769.

Programs

Mathematica

kQ[n_, p_] := Module[{ans = False}, Do[If[Divisible[n - k, p - k], ans = True; Break[]], {k, 1, p - 2}]; ans]; aQ[n_] := EvenQ[n] && Length[(p = FactorInteger[ n][[2 ;; -1, 1]])] > 0 && AllTrue[p, kQ[n, #] &]; Select[Range[500], aQ] (* Amiram Eldar, Aug 17 2019 *)

PARI

getk(p, m) = {for (k=1, p-2, if (((m-k) % (p-k)) == 0, return(k)););}
isok(m) = {if ((m % 2) == 0, my(f = factor(m)[,1]~); if (#f == 1, return (0)); for (i=2, #f, if (!getk(f[i], m), return(0));); return (1););} \\ Michel Marcus, Aug 26 2019

Extensions

More terms from Amiram Eldar, Aug 17 2019