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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.

%I A309780 #19 Aug 31 2019 22:27:22
%S A309780 20,28,44,50,52,68,76,80,88,92,104,110,112,116,124,136,148,152,164,
%T A309780 170,172,176,184,188,196,200,208,212,230,232,236,238,242,244,248,268,
%U A309780 272,284,286,290,292,296,304,316,320,322,328,332,338,344,356,364,368,374
%N 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.
%C A309780 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.
%C A309780 From above, 4^k*p is a term for every prime p >= 5 and k >= 1. - _David A. Corneth_, Aug 17 2019
%C A309780 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
%e A309780 20 = 4*5 is a term (k=2 for p=5).
%e A309780 110 = 10*11 = 22*5 is a term (k=8 for p=11 and k=2 for p=5).
%t A309780 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 *)
%o A309780 (PARI) getk(p, m) = {for (k=1, p-2, if (((m-k) % (p-k)) == 0, return(k)););}
%o A309780 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
%Y A309780 Cf. A309769.
%K A309780 nonn
%O A309780 1,1
%A A309780 _David James Sycamore_, Aug 17 2019
%E A309780 More terms from _Amiram Eldar_, Aug 17 2019