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.

20, 28, 42, 44, 50, 52, 66, 68, 70, 76, 78, 80, 88, 92, 102, 104, 110, 112, 114, 116, 124, 130, 136, 138, 140, 148, 152, 154, 156, 164, 170, 172, 174, 176, 182, 184, 186, 188, 190, 196, 200, 204, 208, 212, 222, 228, 230, 232, 236, 238, 242, 244, 246, 248, 252

Offset: 1

Author

David James Sycamore, Aug 16 2019

Keywords

• nonn

Comments

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).

Examples

			20 = 4*5 is a term because with k=2, 5-k|20-k.
66 = 6*11 is a term (k=6), although when expressed as 66=22*3 no k exists.
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).
		

Crossrefs

Cf. A005843, A309239, A309155, A307390, A309780, A309781.

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 && AnyTrue[p, kQ[n, #] &]; Select[Range[252], 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(1));););} \\ Michel Marcus, Aug 26 2019