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
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]));
(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
KEYWORD
nonn
AUTHOR
Yasutoshi Kohmoto, Feb 24 2007
EXTENSIONS
Edited and extended by R. J. Mathar, Jul 10 2009
STATUS
approved