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

1,2

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

Links

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A093890, A093892.

Cf. A005153.

Sequence in context: A114871 A085150 A051178 * A213708 A239063 A151999

Adjacent sequences: A093888 A093889 A093890 * A093892 A093893 A093894

Keyword

nonn

Author

Amarnath Murthy, Apr 23 2004

Extensions

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

Status

approved