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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 17 09:09 EDT 2019. Contains 325099 sequences. (Running on oeis4.)