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

%I A126849 #19 Nov 17 2017 20:34:14
%S A126849 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,
%T A126849 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,
%U A126849 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
%N A126849 Sum over the divisors d of n constrained to cases where all exponents of the prime factorization of d are prime.
%C A126849 The case a(1) = 1 is set by convention.
%C A126849 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.
%H A126849 Antti Karttunen, <a href="/A126849/b126849.txt">Table of n, a(n) for n = 1..16384</a>
%H A126849 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A126849 sum_{d|n, d=product p_j^r_j, all r_j prime} d.
%F A126849 a(1) = 1, and for n > 1, a(n) = Sum_{d|n, d>1} A293449(d)*d. - _Antti Karttunen_, Nov 17 2017
%e A126849 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.
%e A126849 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.
%t A126849 Array[DivisorSum[#, # &, AllTrue[FactorInteger[#][[All, -1]], PrimeQ] &] &, 96] (* _Michael De Vlieger_, Nov 17 2017 *)
%o A126849 (PARI)
%o A126849 vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
%o A126849 A293449(n) = vecproduct(apply(e -> isprime(e), factorint(n)[, 2]));
%o A126849 A126849(n) = if(1==n,n,sumdiv(n,d,(d>1)*A293449(d)*d)); \\ _Antti Karttunen_, Nov 17 2017
%o A126849 (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
%Y A126849 Cf. A095683, A095691, A096290, A293449.
%K A126849 nonn
%O A126849 1,4
%A A126849 _Yasutoshi Kohmoto_, Feb 24 2007
%E A126849 Edited and extended by _R. J. Mathar_, Jul 10 2009