

A093890


Number of primes arising as the sum of one or more divisors of n.


7



0, 2, 1, 4, 1, 5, 1, 6, 2, 7, 1, 9, 1, 5, 4, 11, 1, 12, 1, 13, 5, 5, 1, 17, 2, 5, 4, 16, 1, 20, 1, 18, 4, 6, 6, 24, 1, 5, 5, 24, 1, 24, 1, 18, 11, 5, 1, 30, 1, 15, 3, 18, 1, 30, 6, 30, 5, 7, 1, 39, 1, 3, 18, 31, 6, 34, 1, 16, 3, 34, 1, 44, 1, 4, 13, 16, 4, 39, 1, 42, 5, 5, 1, 48, 5, 5, 2, 41, 1, 51, 2
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OFFSET

1,2


COMMENTS

a(2^n) = pi(2^(n+1)1).
Except for n=3 and n=42, it appears that the records occur at the highly abundant numbers A002093. The record values appear to be pi(sigma(n)) for n in A002093, which means that these n are members of A093891. [T. D. Noe, Mar 19 2010]


LINKS



EXAMPLE

a(4) = 4, the divisors of 4 are 1, 2 and 4.
Primes arising are 2, 3 = 1 + 2, 5 = 1 + 4 and 7 = 1 + 2 + 4.


MATHEMATICA

Do[l = Subsets[Divisors[n]]; l = Union[Map[Plus @@ #&, l]]; Print[Length[Select[l, PrimeQ]]], {n, 100}] (* Ryan Propper, Jun 04 2006 *)
CountPrimes[n_] := Module[{d=Divisors[n], t, lim, x}, t=CoefficientList[Product[1+x^i, {i, d}], x]; lim=PrimePi[Length[t]1]; Count[t[[1+Prime[Range[lim]]]], _?(#>0 &)]]; Table[CountPrimes[n], {n, 100}] (* T. D. Noe, Mar 19 2010 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



