A375032 -id:A375032 - cp's OEIS frontend

A375033 The maximum even exponent in the prime factorization of n, or 0 if no such exponent exists.

0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Jul 28 2024

First differs from A350386 at n = 36.
The asymptotic density of the occurrences of 0's is d(0) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463; the asymptotic density of the exponentially odd numbers, A268335).
The asymptotic density of the occurrences of 2*k, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+1)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k-1)*(p+1))).
For example, the asymptotic density of the occurrences of 2's is d(1) = Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = 0.243291... (the asymptotic density of A375031).

Cf. A051903, A065463, A162642, A268335, A350386, A350388, A375031, A375032.

a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], EvenQ]]]; a[1] = 0; Array[a, 100]

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

max(a(n), A375032(n)) = A051903(n).
a(n) = 0 if and only if n is an exponentially odd number (A268335).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (2*k) * d(k) = 0.72584606502990528747..., where d(k) is defined in the Comments section above.
a(n) = A051903(A350388(n)). - Amiram Eldar, Aug 17 2024

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

Original entry on oeis.org

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

Amiram Eldar, Aug 31 2024

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).

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

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

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

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

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

a(n) = A375032(n!).

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

A375034 The difference between the maximum odd exponent and the maximum even exponent in the prime factorization of n, where 0 is assigned to each maximum exponent if no such exponent exists.

Original entry on oeis.org

0, 1, 1, -2, 1, 1, 1, 3, -2, 1, 1, -1, 1, 1, 1, -4, 1, -1, 1, -1, 1, 1, 1, 3, -2, 1, 3, -1, 1, 1, 1, 5, 1, 1, 1, -2, 1, 1, 1, 3, 1, 1, 1, -1, -1, 1, 1, -3, -2, -1, 1, -1, 1, 3, 1, 3, 1, 1, 1, -1, 1, 1, -1, -6, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -3

Offset: 1

Amiram Eldar, Jul 28 2024

The indices of high value records are 1, 2, 8, 32, 128, 512, ... (A081294 with offset 1), and the indices of low value records are 1, 4, 16, 64, 256, 1024, ... (A000302 with offset 1).

Cf. A000302, A081294, A368714, A375032, A375033.

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[0, Max[Select[e, OddQ]]] - Max[0, Max[Select[e, EvenQ]]]]; a[1] = 0; Array[a, 100]

a(n) = {my(e = factor(n)[,2], e1 = select(x -> (x % 2), e), e2 = select(x -> !(x % 2), e)); if(#e1 == 0, 0, vecmax(e1)) - if(#e2 == 0, 0, vecmax(e2));}

a(n) = A375032(n) - A375033(n).
a(n) = 0 if and only if n = 1.
a(n) <= 0 if and only if n is in A368714.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (-1)^(k+1)*k*d(k) = 0.5741591604302832339078..., where d(k) = Product_{p prime} (1 - 1/(p^(k+1)*(p+1))) - Product_{p prime} (1 - 1/(p^(k-1)*(p+1))) for k >= 2, and d(1) = Product_{p prime} (1 - 1/(p^2*(p+1))).

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A375033 The maximum even exponent in the prime factorization of n, or 0 if no such exponent exists.

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

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A375034 The difference between the maximum odd exponent and the maximum even exponent in the prime factorization of n, where 0 is assigned to each maximum exponent if no such exponent exists.

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Mathematica

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