A372602 -id:A372602 - 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... .

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... .

A372604 The maximal exponent in the prime factorization of the largest divisor of n whose number of divisors is a power of 2.

Original entry on oeis.org

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, 3, 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, 3, 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 = 32.
Differs from A368247 at n = 1, 128, 216, 256, 384, 432, 512, ... .
All the terms are of the form 2^k-1 (A000225).

			4 has 3 divisors, 1, 2 and 4. The number of divisors of 4 is 3, which is not a power of 2. The number of divisors of 2 is 2, which is a power of 2. Therefore, A372379(4) = 2 and a(4) = A051903(2) = 1.

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

Cf. A000225, A051903, A092323, A372379, A372466.
Cf. A331273, A368247.
Similar sequences: A007424, A368781, A372601, A372602, A372603.

f[n_] := 2^Floor[Log2[n + 1]] - 1; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

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

a(n) = A051903(A372379(n)).
a(n) = A092323(A051903(n)+1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{i>=1} 2^i * (1 - 1/zeta(2^(i+1)-1)) = 1.36955053734097783559... .

A375359 The maximum exponent in the prime factorization of the smallest number whose square is divisible by n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Aug 13 2024

Differs from A050361 at n = 1, 64, 128, 192, ... . Differs from A366902 at n = 1, 64, 192, 216, ... . Differs from A325837 at n = 1, 216, 432, 648, ... .

Cf. A019554, A051903, A372602, A375360.

Cf. A050361, A325837, A366902.

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[(If[EvenQ[#], #, # + 1]) & /@ e]/2]; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, vecmax(apply(x -> if(x % 2, x+1, x), factor(n)[,2]))/2);

a(n) = A051903(A019554(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum{k>=1} (1 - 1/zeta(2*k+1)) = 1.21464720975357037829... .

cp's OEIS Frontend

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

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

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A372604 The maximal exponent in the prime factorization of the largest divisor of n whose number of divisors is a power of 2.

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A375359 The maximum exponent in the prime factorization of the smallest number whose square is divisible by n.

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