A372604 -id:A372604 - cp's OEIS frontend

A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 07 2024

First differs from A331273 at n = 64.

Differs from A363332 at n = 1, 216, 432, 648, 864, 1000, ... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051903, A109613, A350390, A368711.
Cf. A331273, A363332.
Similar sequences: A007424, A368781, A372602, A372603, A372604.

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

s(n) = (n+1) \ 2 * 2 - 1;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A350390(n)).
a(n) = A109613(A051903(n)-1) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + 2 * Sum_{i>=1} (1 - (1/zeta(2*i+1))) = 1.42929441950714075659... .

A372602 The maximal exponent in the prime factorization of the largest square dividing n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 07 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008833, A051903, A052928.

Similar sequences: A007424, A368781, A372601, A372603, A372604.

f[n_] := 2 * Floor[n/2]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = n \ 2 * 2;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A008833(n)).
a(n) = A052928(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * Sum_{i>=1} (1 - (1/zeta(2*i))) = 0.98112786070359477197... .

A372603 The maximal exponent in the prime factorization of the powerful part of n.

Original entry on oeis.org

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, 2, 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, 3, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0

Offset: 1

Amiram Eldar, May 07 2024

First differs from A275812 at n = 36, and from A212172 at n = 37.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051903, A057521, A087156, A368710.
Cf. A013661, A033150, A059956.
Cf. A212172, A275812.
Similar sequences: A007424, A368781, A372601, A372602, A372604.

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

s(n) = if(n == 1, 0, n);
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A057521(n)).
a(n) = A087156(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - 1/zeta(2) + Sum_{i>=2} (1 - 1/zeta(i)) = A033150 - A059956 = 1.09728403825134113562... .

cp's OEIS Frontend

A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A372602 The maximal exponent in the prime factorization of the largest square dividing n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A372603 The maximal exponent in the prime factorization of the powerful part of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula