

A093891


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


6



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
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OFFSET

1,2


COMMENTS

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



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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



