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A275812
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Sum of exponents larger than one in the prime factorization of n: A001222(n) - A056169(n).
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16
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0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 2, 4, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 3
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OFFSET
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1,4
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LINKS
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FORMULA
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Additive with a(p) = 0, and a(p^e) = e for e >= 2.
a(n) >= 0, with equality if and only if n is squarefree (A005117).
a(n) <= A001222(n), with equality if and only if n is powerful (A001694).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (1/p^2 + 1/(p*(p-1))) = A085548 + A136141 = 1.22540408909086062637... . (End)
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MATHEMATICA
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Table[Total@ Map[Last, Select[FactorInteger@ n, Last@ # > 1 &] /. {} -> {{0, 0}}], {n, 120}] (* Michael De Vlieger, Aug 11 2016 *)
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PROG
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(Scheme, two variants, the first one with memoizing definec-macro)
(Perl) sub a275812 { vecsum( grep {$_> 1} map {$_->[1]} factor_exp(shift) ); } # Dana Jacobsen, Aug 15 2016
(Python)
from sympy import factorint, primefactors
def a001222(n):
return 0 if n==1 else a001222(n//primefactors(n)[0]) + 1
def a056169(n):
f=factorint(n)
return 0 if n==1 else sum(1 for i in f if f[i]==1)
def a(n):
return a001222(n) - a056169(n)
(PARI) a(n) = my(f = factor(n)); sum(k=1, #f~, if (f[k, 2] > 1, f[k, 2])); \\ Michel Marcus, Jul 19 2017
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CROSSREFS
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Differs from A212172 for the first time at n=36, where a(36)=4, while A212172(36)=2.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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