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

Author

T. D. Noe, Jun 17 2009

Keywords

• nonn

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.

Examples

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

Links

• Felix Huber, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Programs

Maple

with(NumberTheory):
A161510:=proc(n)
    local b,l,j;
    l:=[(Divisors(n))[]]:
    b:=proc(m,i)
        option remember;
        `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
    end;
    add(b(ithprime(j),nops(l)),j=1..pi(sigma(n)));
end:
seq(A161510(n),n=1..77); # Felix Huber, Jul 24 2025

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}]

PARI

a(n) = my(nb=0, d=divisors(n)); forsubset(#d, s, nb+=isprime(sum(i=1, #s, d[s[i]]))); nb; \\ Michel Marcus, Jul 24 2025