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A008474
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If n = Product (p_j^k_j) then a(n) = Sum (p_j + k_j).
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20
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0, 3, 4, 4, 6, 7, 8, 5, 5, 9, 12, 8, 14, 11, 10, 6, 18, 8, 20, 10, 12, 15, 24, 9, 7, 17, 6, 12, 30, 13, 32, 7, 16, 21, 14, 9, 38, 23, 18, 11, 42, 15, 44, 16, 11, 27, 48, 10, 9, 10, 22, 18, 54, 9, 18, 13, 24, 33, 60, 14, 62, 35, 13, 8, 20, 19, 68, 22, 28, 17, 72, 10, 74, 41, 11, 24, 20, 21
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OFFSET
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1,2
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COMMENTS
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a(1) = 0 by convention, but could equally be taken to be 1 or 2.
Since only the primes p_j with nonzero exponents k_j in the factorization of n are considered in Sum (p_j + k_j), to the empty product (1) should correspond the empty sum (0). a(1) = 0 is thus the most natural choice. - Daniel Forgues, Apr 06 2010
Conjecture: for m > 4, by iterating the map m -> A008474(m) one always reaches 5 [tested up to m = 320000]. - Ivan N. Ianakiev, Nov 10 2014
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LINKS
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FORMULA
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Additive with a(p^e) = p + e.
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MAPLE
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A008474 := proc(n) local e, j; e := ifactors(n)[2]:
add(e[j][1]+e[j][2], j=1..nops(e)) end:
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MATHEMATICA
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PROG
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(Haskell)
a008474 n = sum $ zipWith (+) (a027748_row n) (a124010_row n)
(PARI) {for(k=1, 79,
M=factor(k); smt =0;
for(i=1, matsize(M)[1], for(j=1, matsize(M)[2], smt=smt+M[i, j]));
(Python)
from sympy import factorint
def a(n): return 0 if n == 1 else sum(p+k for p, k in factorint(n).items())
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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