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The maximum odd exponent in the prime factorization of n!.
2

%I #11 Aug 31 2024 18:00:58

%S 1,1,3,3,1,1,7,7,1,1,5,5,11,11,15,15,3,3,1,9,19,19,3,3,23,23,25,25,7,

%T 7,31,31,15,15,17,17,35,35,9,9,39,39,41,41,21,21,3,3,47,47,49,49,3,13,

%U 53,53,27,27,9,9,57,57,63,63,31,31,31,15,67,67,11,11

%N The maximum odd exponent in the prime factorization of n!.

%C The sequence of indices of record values, 2, 4, 8, 14, 16, 22, 26, 28, 32, 38, ..., are the odious numbers (A000069) multiplied by 2 (A128309).

%H Amiram Eldar, <a href="/A375849/b375849.txt">Table of n, a(n) for n = 2..10000</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%F a(n) = A375032(n!).

%F max(a(n), A375850(n)) = A011371(n).

%t a[n_] := Max[Select[FactorInteger[n!][[;; , 2]], OddQ]]; Array[a, 100, 2]

%o (PARI) a(n) = {my(e = select(x -> (x % 2), factor(n!)[, 2])); if(#e > 0, vecmax(e));}

%o (Python)

%o from collections import Counter

%o from sympy import factorint

%o def A375849(n): return max(filter(lambda x: x&1,sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values())) # _Chai Wah Wu_, Aug 31 2024

%Y Cf. A000069, A000142, A011371, A128309, A375032, A375850.

%K nonn,easy

%O 2,3

%A _Amiram Eldar_, Aug 31 2024