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

1,4

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences related to sums of divisors

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A095683, A095691, A096290, A293449.

Sequence in context: A239261 A242707 A236379 * A284117 A183099 A162296

Adjacent sequences: A126846 A126847 A126848 * A126850 A126851 A126852

Keyword

nonn

Author

Yasutoshi Kohmoto, Feb 24 2007

Extensions

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

Status

approved