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A087436 Number of odd prime factors of n, counted with repetitions.

%I A087436 #45 May 16 2025 08:51:17
%S A087436 0,0,1,0,1,1,1,0,2,1,1,1,1,1,2,0,1,2,1,1,2,1,1,1,2,1,3,1,1,2,1,0,2,1,
%T A087436 2,2,1,1,2,1,1,2,1,1,3,1,1,1,2,2,2,1,1,3,2,1,2,1,1,2,1,1,3,0,2,2,1,1,
%U A087436 2,2,1,2,1,1,3,1,2,2,1,1,4,1,1,2,2,1,2,1,1,3,2,1,2,1,2,1,1,2,3,2,1,2
%N A087436 Number of odd prime factors of n, counted with repetitions.
%C A087436 Number of parts larger than 1 in the partition with Heinz number n. The Heinz number of an integer partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by _Alois P. Heinz_ in A215366 as an "encoding" of a partition). Example: a(9) = 2 because the partition with Heinz number 9 (=3*3) is [2,2]. - _Emeric Deutsch_, Oct 02 2015
%C A087436 Totally additive because both A001222 and A007814 are. a(2) = 0, and a(p) = 1 for odd primes p, a(m*n) = a(m)+a(n) for m, n > 1. - _Antti Karttunen_, Jul 10 2020
%H A087436 Antti Karttunen, <a href="/A087436/b087436.txt">Table of n, a(n) for n = 1..65537</a> (first 1000 terms from G. C. Greubel)
%F A087436 a(n) = A001222(n) - A007814(n).
%F A087436 a(n) = A001222(A000265(n)). - _Antti Karttunen_, Jul 10 2020
%F A087436 Sum_{k=1..n} a(k) = n * (log(log(n)) + B_2 - 1) + O(n/log(n)), where B_2 = A083342. - _Amiram Eldar_, May 16 2025
%e A087436 a(9) = 2 because 9 = 3*3 has 2 odd prime factors. - _Emeric Deutsch_, Oct 02 2015
%p A087436 seq(bigomega(n) - padic[ordp](n, 2), n=1..102); # _Peter Luschny_, Dec 06 2017
%t A087436 Join[{0},Table[Length[Select[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ n]],OddQ]],{n,2,110}]] (* _Harvey P. Dale_, Feb 01 2013 *)
%o A087436 (PARI) a(n) = bigomega(n) - valuation(n, 2); \\ _Michel Marcus_, Sep 10 2019
%o A087436 (PARI) A087436(n) = (bigomega(n>>valuation(n,2))); \\ _Antti Karttunen_, Jul 10 2020
%Y A087436 Cf. A000244 (the first occurrence of each n, and also the positions of records).
%Y A087436 Cf. A000265, A001222, A005087, A007814, A083342, A215366, A336118, A336161.
%K A087436 nonn,easy
%O A087436 1,9
%A A087436 _Reinhard Zumkeller_, Sep 03 2003