A375849 The maximum odd exponent in the prime factorization of n!.

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

Offset: 2

Author

Amiram Eldar, Aug 31 2024

Keywords

• nonn
• easy

Comments

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).

Links

• Amiram Eldar, Table of n, a(n) for n = 2..10000
• Index entries for sequences related to factorial numbers.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Python

from collections import Counter
from sympy import factorint
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

Formula

a(n) = A375032(n!).

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