A093894 -id:A093894 - cp's OEIS frontend

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

Amarnath Murthy, Apr 23 2004

nonn

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

Cf. A093890, A093891, A093892, A093894.

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 *)

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

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

A160994 a(n) is the least prime number p such that every sum of two or more divisors of p^n is composite.

Original entry on oeis.org

3, 7, 7, 19, 19, 139, 151, 211, 211, 211, 421, 2311, 2311, 92401, 120121, 120121, 180181, 2312311

Offset: 1

Robert G. Wilson v, Jun 01 2009

Cf. A093893, A093894.

(* first do *) Needs["Combinatorica`"] (* then *) f[n_] := Block[{d = Divisors@n, k, mx}, k = 1 + Length@d; mx = 2^Length[d]; While[k < mx && !PrimeQ[Plus @@ NthSubset[k, d]], k++ ]; If[k == mx, Length@d, 0]];
a[n_] := a[n] = Module[{p = If[n == 1, 2, a[n-1]]}, While[f[p^n] == 0, p = NextPrime[p]]; p]; Array[a, 13] (* second part of the program added by Amiram Eldar, Jul 30 2024 *)

Offset changed to 1 and name corrected by Amiram Eldar, Jul 30 2024

cp's OEIS Frontend

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

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