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Sum of prime factors of sigma(n).
1

%I #23 May 02 2021 11:41:01

%S 3,2,7,5,5,2,8,13,5,5,9,9,5,5,31,5,16,7,12,2,5,5,10,31,12,7,9,10,5,2,

%T 10,5,5,5,20,21,10,9,10,12,5,13,12,18,5,5,33,22,34,5,9,5,10,5,10,7,10,

%U 10,12,33,5,15,127,12,5,19,12,5,5,5,21,39,24,33,14,5,12,7,36,11,12,12,9,5

%N Sum of prime factors of sigma(n).

%H Indranil Ghosh, <a href="/A080649/b080649.txt">Table of n, a(n) for n = 2..10000</a>

%F a(n) = A008472(A000203(n)). - _Indranil Ghosh_, Jan 13 2017

%e For n = 10, sigma(10) = 18 and the sum of the prime factors of 18 is 2 + 3 = 5. So, a(10) = 5. - _Indranil Ghosh_, Jan 13 2017

%t Table[Apply[Plus, Transpose[FactorInteger[DivisorSigma[1, n]]][[1]]], {n, 3, 100}]

%o (Python)

%o from sympy import isprime

%o def sigma(n):

%o return sum(i for i in range(1,n+1) if n % i == 0)

%o def sopf(n):

%o return sum(i for i in range(1,n+1) if n % i == 0 and isprime(i))

%o def A080649(n):

%o return sopf(sigma(n))

%o for i in range(2,101):

%o print(str(i)+" "+str(A080649(i))) # _Indranil Ghosh_, Jan 15 2017

%o (PARI) a(n) = vecsum(factor(sigma(n))[,1]); \\ _Michel Marcus_, Jan 14 2017

%Y Cf. A000203, A008472.

%K nonn

%O 2,1

%A _Joseph L. Pe_, Feb 28 2003