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A309769 Even numbers m having at least one odd prime divisor p for which there exists a positive integer k < p-1 such that p-k|m-k.

%I A309769 #23 Aug 31 2019 22:48:37
%S A309769 20,28,42,44,50,52,66,68,70,76,78,80,88,92,102,104,110,112,114,116,
%T A309769 124,130,136,138,140,148,152,154,156,164,170,172,174,176,182,184,186,
%U A309769 188,190,196,200,204,208,212,222,228,230,232,236,238,242,244,246,248,252
%N A309769 Even numbers m having at least one odd prime divisor p for which there exists a positive integer k < p-1 such that p-k|m-k.
%C A309769 Complement in A005843 of A309239. Every odd number > 1 has the property mentioned in Name, but these are the only even numbers with this property. No term is either a power of 2 or a semiprime. A number m is a term if and only if m = 2rp, where r >= 2, and p is a prime > q, the smallest prime divisor of 2r-1 (k=p-q). For any given r, 2rz is the smallest multiple of 2r in this sequence, where z=nextprime(q). If m = 2rp is a term and 2r-1 is prime, then p is the greatest prime divisor of m (the converse is not true; e.g., m=70=10*7).
%e A309769 20 = 4*5 is a term because with k=2, 5-k|20-k.
%e A309769 66 = 6*11 is a term (k=6), although when expressed as 66=22*3 no k exists.
%e A309769 110 = 10*11 = 22*5 is a term for two reasons, since with both of its odd prime factors it has the required property; 5-2|110-2 and 11-8|110-8. This is the smallest term having two distinct odd prime factors, both of which have the above property (see A309780, A309781).
%t A309769 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 && AnyTrue[p, kQ[n, #] &]; Select[Range[252], aQ] (* _Amiram Eldar_, Aug 17 2019 *)
%o A309769 (PARI) getk(p, m) = {for (k=1, p-2, if (((m-k) % (p-k)) == 0, return(k)););}
%o A309769 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(1));););} \\ _Michel Marcus_, Aug 26 2019
%Y A309769 Cf. A005843, A309239, A309155, A307390, A309780, A309781.
%K A309769 nonn
%O A309769 1,1
%A A309769 _David James Sycamore_, Aug 16 2019