A275812 Sum of exponents larger than one in the prime factorization of n: A001222(n) - A056169(n).

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

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

Formula

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)).

a(n) = A001222(n) - A056169(n).

a(n) = A001222(A057521(n)). - Antti Karttunen, Jul 19 2017

From Amiram Eldar, Sep 28 2023: (Start)

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)

a(n) = A046660(n) + A056170(n). - Amiram Eldar, Jan 09 2024

Mathematica

Table[Total@ Map[Last, Select[FactorInteger@ n, Last@ # > 1 &] /. {} -> {{0, 0}}], {n, 120}] (* Michael De Vlieger, Aug 11 2016 *)

Prog

(Scheme, two variants, the first one with memoizing definec-macro)
(definec (A275812 n) (if (= 1 n) 0 (+ (if (> (A067029 n) 1) (A067029 n) 0) (A275812 (A028234 n)))))
(define (A275812 n) (- (A001222 n) (A056169 n)))
(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)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 19 2017
(PARI) a(n) = my(f = factor(n)); sum(k=1, #f~, if (f[k, 2] > 1, f[k, 2])); \\ Michel Marcus, Jul 19 2017

Crossrefs

Cf. A001222, A001694, A005117, A028234, A046660, A056169, A056170, A057521, A067029, A247180.

Differs from A212172 for the first time at n=36, where a(36)=4, while A212172(36)=2.

Cf. A085548, A136141.

Sequence in context: A375339 A372603 A212172 * A280683 A171871 A076260

Adjacent sequences: A275809 A275810 A275811 * A275813 A275814 A275815

Keyword

nonn,easy

Author

Antti Karttunen, Aug 11 2016

Status

approved