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%I #34 Jan 09 2024 08:47:04
%S 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,
%T 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,
%U 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
%N Sum of exponents larger than one in the prime factorization of n: A001222(n) - A056169(n).
%H Antti Karttunen, <a href="/A275812/b275812.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F a(1) = 0, and for n > 1, if A067029(n)=1 [when n is one of the terms of A247180], a(n) = a(A028234(n)), otherwise a(n) = A067029(n)+a(A028234(n)).
%F a(n) = A001222(n) - A056169(n).
%F a(n) = A001222(A057521(n)). - _Antti Karttunen_, Jul 19 2017
%F From _Amiram Eldar_, Sep 28 2023: (Start)
%F Additive with a(p) = 0, and a(p^e) = e for e >= 2.
%F a(n) >= 0, with equality if and only if n is squarefree (A005117).
%F a(n) <= A001222(n), with equality if and only if n is powerful (A001694).
%F 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)
%F a(n) = A046660(n) + A056170(n). - _Amiram Eldar_, Jan 09 2024
%t Table[Total@ Map[Last, Select[FactorInteger@ n, Last@ # > 1 &] /. {} -> {{0, 0}}], {n, 120}] (* _Michael De Vlieger_, Aug 11 2016 *)
%o (Scheme, two variants, the first one with memoizing definec-macro)
%o (definec (A275812 n) (if (= 1 n) 0 (+ (if (> (A067029 n) 1) (A067029 n) 0) (A275812 (A028234 n)))))
%o (define (A275812 n) (- (A001222 n) (A056169 n)))
%o (Perl) sub a275812 { vecsum( grep {$_> 1} map {$_->[1]} factor_exp(shift) ); } # _Dana Jacobsen_, Aug 15 2016
%o (Python)
%o from sympy import factorint, primefactors
%o def a001222(n):
%o return 0 if n==1 else a001222(n//primefactors(n)[0]) + 1
%o def a056169(n):
%o f=factorint(n)
%o return 0 if n==1 else sum(1 for i in f if f[i]==1)
%o def a(n):
%o return a001222(n) - a056169(n)
%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jun 19 2017
%o (PARI) a(n) = my(f = factor(n)); sum(k=1, #f~, if (f[k,2] > 1, f[k,2])); \\ _Michel Marcus_, Jul 19 2017
%Y Cf. A001222, A001694, A005117, A028234, A046660, A056169, A056170, A057521, A067029, A247180.
%Y Differs from A212172 for the first time at n=36, where a(36)=4, while A212172(36)=2.
%Y Cf. A085548, A136141.
%K nonn,easy
%O 1,4
%A _Antti Karttunen_, Aug 11 2016
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