A093893 Numbers n such that every sum of two or more divisors is composite.

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 183, 191, 193, 197, 199, 211, 213, 217, 223, 227, 229, 233

Offset: 1

Author

Amarnath Murthy, Apr 23 2004

Keywords

• nonn

Comments

All terms are odd and very few are composite. Every odd prime is a trivial member.

Very few terms have more than four divisors. The smallest such term is 4753, which has six divisors: 1,7,49,97,679,4753. - Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004

Crossrefs

Cf. A093890, A093891, A093892, A093894.

Programs

Mathematica

For[a:=3, a<=500, s =Divisors[a];n := 1;d := False; While[(n<=2^Length[s])\[And]( ["not" character]d), If[Length[NthSubset[n, s]]>=2, If[ !PrimeQ[Plus@@NthSubset[n, s]], n++, d:= True], n++ ]]; If[ ["not" character]d, Print[a]];a+=2]; (Kalman)
fQ[n_] := Union@ PrimeQ[Plus @@@ Subsets[ Divisors@n, {2, Infinity}]] == {False}; Select[ Range[3, 235, 2], fQ@# &] (* Robert G. Wilson v, May 25 2009 *)

Extensions

More terms from Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004

a(1)=1 prepended by Max Alekseyev, Mar 31 2015