A375933 -id:A375933 - cp's OEIS frontend

A375932 The largest unitary k-free divisor of n where k = A051903(n) is the maximum exponent in the prime factorization of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 11, 5, 1, 1, 3, 1, 2, 1, 13, 1, 2, 1, 7, 1, 1, 1, 15, 1, 1, 7, 1, 1, 1, 1, 17, 1, 1, 1, 9, 1, 1, 3, 19, 1, 1, 1, 5, 1, 1, 1, 21, 1

Offset: 1

Amiram Eldar, Sep 03 2024

The product of the prime powers in the prime factorization of n that have an exponent that is smaller than the maximum exponent in this factorization.

			60 = 2^2 * 3 * 5, and the maximum exponent in the prime factorization of 60 is 2, which is the exponent of its prime factor 2. Therefore a(60) = 3 * 5 = 15.

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

Cf. A005117, A049606, A051903, A072774, A375931, A375933.

a[n_] := Module[{f = FactorInteger[n], p, e, i, m}, p = f[[;; , 1]]; e = f[[;; , 2]]; m = Max[e]; i = Position[e, m] // Flatten; n / (Times @@ p[[i]])^m]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2], m); if(n == 1, 1, m = vecmax(e); prod(i = 1, #p, if(e[i] < m, p[i]^e[i], 1)));}

If n = Product_{i} p_i^e_i (where p_i are distinct primes) then a(n) = Product_{i} p_i^(e_i * [e_i < max_{j} e_j]), where [] is the Iverson bracket.
a(n) = n / A375931(n).
a(n) = 1 if and only if n is a power of a squarefree number (A072774).
A051903(a(n)) = A375933(n).
a(n!) = A049606(n) for n != 3.

A375934 Numbers whose prime factorization has a second-largest exponent that equals 1.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 204

Offset: 1

Amiram Eldar, Sep 03 2024

First differs from A332785 at n = 112: A332785(112) = 360 = 2^3 * 3^2 * 5 is not a term of this sequence.
First differs from A317616 at n = 38: A317616(38) = 144 = 2*4 * 3^2 is not a term of this sequence.
Numbers k such that A375933(k) = 1.
Numbers of the form s1 * s2^e, where s1 and s2 are coprime squarefree numbers that are both larger than 1, and e >= 2.
The asymptotic density of this sequence is Sum_{e>=2} d(e) = 0.36113984820338109927..., where d(e) = Product_{p prime} (1 - 1/p^2 + 1/p^e - 1/p^(e+1)) - Product_{p prime} (1 - 1/p^(e+1)) is the asymptotic density of terms k with A051903(k) = e >= 2.

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

Equals A375142 \ A072777.
Subsequence of A013929.
Subsequences: A054753, A065036, A072357, A095990, A096156, A178739, A178740, A179664, A179668, A179692, A189987, A360793, A375076.
Cf. A001221, A005117, A051903, A051904, A072777, A317616, A332785, A375142, A375933.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[0, Max[Select[e, # < Max[e] &]]] == 1]; Select[Range[300], q]

is(n) = if(n == 1, 0, my(e = factor(n)[,2]); e = select(x -> x < vecmax(e), e); if(#e == 0, 0, vecmax(e) == 1));

A051904(a(n)) = 1.
A051903(a(n)) >= 2.
A001221(a(n)) = 2.

A376251 Numbers that have a second-largest exponent in their prime factorization and it is smaller by 1 than the largest exponent.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 180, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 252, 260, 261, 268, 275, 276, 279

Offset: 1

Amiram Eldar, Sep 17 2024

First differs from its subsequence A325241 at n = 74: a(74) = 360 = 2^3 * 3^2 * 5 is not a term of A325241.
Numbers k such that 0 < A375933(k) = A051903(k) - 1.
The asymptotic density of this sequence is Sum_{k>=2} d(k) = 0.24179287499021146826..., where d(2) = 1/zeta(3) - 1/zeta(2), and d(k) = 1/zeta(k+1) - 1/zeta(k) + 1/zeta(k-1) - Product_{p prime} (1 - 1/p^(k-1) + 1/p^k - 1/p^(k+1)) for k >= 3.

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

Subsequence of A013929.
Subsequences: A067259, A325241, A376249.
Cf. A002117, A013661, A051903, A375933.

q[k_] := Module[{e = Union[FactorInteger[k][[;; , 2]]]}, Length[e] > 1 && e[[-2]] + 1 == e[[-1]]]; Select[Range[300], q]

is(k) = {my(e = Set(factor(k)[, 2])); #e > 1 && e[#e-1] + 1 == e[#e]; }

cp's OEIS Frontend

A375932 The largest unitary k-free divisor of n where k = A051903(n) is the maximum exponent in the prime factorization of n.

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A375934 Numbers whose prime factorization has a second-largest exponent that equals 1.

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A376251 Numbers that have a second-largest exponent in their prime factorization and it is smaller by 1 than the largest exponent.

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Mathematica

PARI