A126849 Sum over the divisors d of n constrained to cases where all exponents of the prime factorization of d are prime.

1, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 4, 0, 0, 0, 12, 0, 9, 0, 4, 0, 0, 0, 12, 25, 0, 36, 4, 0, 0, 0, 44, 0, 0, 0, 49, 0, 0, 0, 12, 0, 0, 0, 4, 9, 0, 0, 12, 49, 25, 0, 4, 0, 36, 0, 12, 0, 0, 0, 4, 0, 0, 9, 44, 0, 0, 0, 4, 0, 0, 0, 129, 0, 0, 25, 4, 0, 0, 0, 12, 36, 0, 0, 4, 0, 0, 0, 12, 0, 9, 0, 4, 0, 0, 0, 44

Offset: 1

Author

Yasutoshi Kohmoto, Feb 24 2007

Keywords

• nonn

Comments

The case a(1) = 1 is set by convention.

Note that this is different from the PPsigma function defined in A096290, where PPsigma(12)=PPsigma(2^2*3^1)=0 since the factor 3^1 appears with an exponent too small to yield a nonzero sum.

Examples

			a(12) = 2^2 = 4 because 4 is the only divisor of the divisors set 1, 2 = 2^1, 3 = 3^1, 4 = 2^2, 6 = 2^1 * 3^1, 12 = 2^2 * 3^1 for which all the exponents are prime.
a(9) = 9 because 9 is the only divisor of the set 1, 3 = 3^1, 9 = 3^2 for which all the exponents are prime.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A095683, A095691, A096290, A293449.

Programs

Mathematica

Array[DivisorSum[#, # &, AllTrue[FactorInteger[#][[All, -1]], PrimeQ] &] &, 96] (* Michael De Vlieger, Nov 17 2017 *)

PARI

vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A293449(n) = vecproduct(apply(e -> isprime(e), factorint(n)[, 2]));
A126849(n) = if(1==n,n,sumdiv(n,d,(d>1)*A293449(d)*d)); \\ Antti Karttunen, Nov 17 2017

PARI

first(n) = {my(res = vector(n)); res[1] = 1; forprime(p = 2, sqrtint(n), forprime(e = 2, logint(n, p), for(k = 1, n \ (p^e), res[k*p^e] += p^e))); res} \\ David A. Corneth, Nov 17 2017

Formula

sum_{d|n, d=product p_j^r_j, all r_j prime} d.

a(1) = 1, and for n > 1, a(n) = Sum_{d|n, d>1} A293449(d)*d. - Antti Karttunen, Nov 17 2017

Extensions

Edited and extended by R. J. Mathar, Jul 10 2009