A161510 Number of primes formed as the sum of distinct divisors of n, counted with repetition.

0, 2, 1, 4, 1, 6, 1, 6, 2, 7, 1, 20, 1, 5, 4, 11, 1, 16, 1, 19, 5, 5, 1, 66, 2, 5, 4, 17, 1, 64, 1, 18, 4, 6, 6, 120, 1, 5, 5, 63, 1, 62, 1, 18, 11, 5, 1, 237, 1, 15, 3, 18, 1, 47, 6, 60, 5, 7, 1, 863, 1, 3, 20, 31, 6, 58, 1, 16, 3, 62, 1, 808, 1, 4, 13, 16, 4, 56, 1, 216, 5, 5, 1, 839, 5, 5

Offset

1,2

Comments

That is, if a number has d divisors, then we compute all 2^d sums of distinct divisors and count how many primes are formed. Sequence A093893 lists the n that produce no primes except for the primes that divide n. The Mathematica code works well for numbers up to about 221760, which has 168 divisors and creates a polynomial of degree 950976. The coefficients of the prime powers of that polynomial sum to 28719307224839120896278355000770621322645671888269, the number of primes formed by the divisors of 221760. Records appear to occur at n=10 and n in A002182, the highly composite numbers.

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Example

a(4) = 4 because the divisors (1,2,4) produce 4 primes (2,1+2,1+4,1+2+4).

Mathematica

CountPrimes[n_] := Module[{d=Divisors[n], t, lim, x}, t=CoefficientList[Product[1+x^d[[i]], {i, Length[d]}], x]; lim=PrimePi[Length[t]-1]; Plus@@t[[1+Prime[Range[lim]]]]]; Table[CountPrimes[n], {n, 100}]

Crossrefs

Sequence in context: A329638 A322036 A318441 * A329379 A328479 A340346

Adjacent sequences: A161507 A161508 A161509 * A161511 A161512 A161513

Keyword

nonn

Author

T. D. Noe, Jun 17 2009

Status

approved