A093891 Numbers k such that every prime up to sigma(k) is a sum of divisors of k.

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240

Offset: 1

Author

Amarnath Murthy, Apr 23 2004

Keywords

• nonn

Comments

Sequence is infinite as sigma (2^n) = 2^(n+1)-1 and a(2^n) = pi(2^(n+1)-1).

Does this sequence include any non-members of A005153 other than 10, 70 and 836? - Franklin T. Adams-Watters, Apr 28 2006

The answer to the previous comment is yes, this sequence has many terms that are not in A005153. See A174434. - T. D. Noe, Mar 19 2010

Examples

			4 is a member as sigma(4) = 7 and all the primes up to 7 are a partial sum of divisors of 4, since divisors of 4 are 1, 2 and 4 and because primes arising are 2, 3 = 1+2, 5 = 1+4 and 7 = 1+2+4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A093890, A093892.

Cf. A005153.

Programs

Mathematica

Select[Range[240], SubsetQ[Total /@ Rest@ Subsets@ Divisors[#], Prime@ Range@ PrimePi@ DivisorSigma[1, #]] &] (* Michael De Vlieger, Mar 19 2021 *)

PARI

isok(m) = {my(d=divisors(m), vp = primes(primepi(sigma(m)))); for (i=1, 2^#d - 1, my(b = Vecrev(binary(i)), x = sum(k=1, #b, b[k]*d[k])); if (vecsearch(vp, x), vp = setminus(vp, Set(x))); if (#vp == 0, return (1)););} \\ Michel Marcus, Mar 19 2021

Extensions

More terms from Franklin T. Adams-Watters, Apr 28 2006