A368715 -id:A368715 - cp's OEIS frontend

A368714 Numbers whose maximal exponent in their prime factorization is even.

1, 4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207, 208

Offset: 1

Amiram Eldar, Jan 04 2024

First differs from A240112 at n = 30.
Numbers k such that A051903(k) is even.
The asymptotic density of this sequence is Sum_{k>=2} (-1)^k * (1 - 1/zeta(k)) = 0.27591672059822700769... .

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

Cf. A051903, A240112.
Subsequences: A000290, A000302, A000583, A001014, A001016, A001025, A008454, A062503, A067259.
Similar sequences: A060476, A074661, A096432, A336064, A368715.

Select[Range[210], # == 1 || EvenQ[Max[FactorInteger[#][[;;, 2]]]] &]

lista(kmax) = for(k = 1, kmax, if(k == 1 || !(vecmax(factor(k)[,2])%2), print1(k, ", ")));

A381952 a(n) is the greatest common divisor of n and the maximum exponent in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Mar 11 2025

The asymptotic density d(k) of the occurrences of each integer k >= 1 in this sequence can be calculated. E.g., d(1) = 0.75001550800919720296... (1 - the density of A368715), and d(2) = 0.16205634516436945215... (see A381953). From these densities the asymptotic mean of this sequence can be evaluated by Sum_{k>=1} k*d(k), but it seems that the expressions for d(k) for large values of k may be complicated.

The sums of the first 10^k terms, for k = 1, 2, ..., are 11, 135, 1396, 14014, 140241, 1402521, 14025251, 140252636, 1402526282, 14025262924, ... . Apparently, the asymptotic mean of this sequence equals 1.402526... .

			a(1) = gcd(1, A051903(1)) = gcd(1, 0) = 1.
a(4) = gcd(4, A051903(4)) = gcd(4, 2) = 2.
a(16) = gcd(16, A051903(16)) = gcd(16, 4) = 4.

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

Cf. A051903, A336064, A368715, A381953.

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

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

a(n) = gcd(n, A051903(n)).
a(n) >= 2 if and only if n is in A368715.
a(A381953(n)) = 2.
a(A336064(n)) = A051903(A336064(n)).

A374587 The maximum exponent in the prime factorization of the numbers that are not coprime to the maximum exponent in their prime factorization.

Original entry on oeis.org

2, 2, 4, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 3, 2, 6, 2, 3, 2, 4, 2, 2, 2, 2, 2, 3, 4, 2, 3, 2, 2, 2, 3, 2, 4, 2, 2, 2, 5, 4, 2, 3, 2, 4, 2, 2, 3, 6, 2, 2, 2, 4, 2, 3, 2, 2, 2, 2, 4, 2, 2, 2, 8, 2, 3, 2, 3, 4, 2, 2, 2, 2, 3, 2, 4, 2, 2, 3, 2, 6, 4, 2, 4, 2, 2, 2, 2

Offset: 1

Amiram Eldar, Jul 12 2024

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

Cf. A051903, A368715, A374586.

f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[!CoprimeQ[n, e], e, Nothing]]; Array[f, 350]

lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(gcd(k, e) > 1, print1(e, ", ")));}

a(n) = A051903(A368715(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} k * d(k) / Sum_{k>=2} d(k) = 2.74240523513766773312..., where d(k) = (1 - 1/f(k+1, k))/zeta(k+1) - (1 - 1/f(k, k))/zeta(k), and f(e, m) = Product_{primes p|m} ((1-1/p^e)/(1-1/p)).

A381953 Numbers k such that A381952(k) = 2.

Original entry on oeis.org

4, 12, 18, 20, 28, 36, 44, 50, 52, 60, 64, 68, 76, 84, 90, 92, 98, 100, 116, 124, 126, 132, 140, 148, 150, 156, 162, 164, 172, 180, 188, 196, 198, 204, 212, 220, 228, 234, 236, 242, 244, 252, 260, 268, 276, 284, 292, 294, 300, 306, 308, 316, 320, 332, 338, 340

Offset: 1

Amiram Eldar, Mar 11 2025

Disjoint union of {2 * m | m is an odd number such that A051903(m) > 0 and is divisible by 4}, {2^e * m | m is an odd number such that A051903(m) == 2 (mod 4), gcd(A051903(m), m) = 1), 1 <= e <= A051903(m)}, and {2^e * m | A051903(m) < e, e == 2 (mod 4), gcd(e, m) = 1}.

The asymptotic density of this sequence is Sum_{k>=1} (1-2^(4*k)/((2^(4*k)-1)*zeta(4*k)) - (1-2^(4*k+1)/((2^(4*k+1)-1)*zeta(4*k+1)))))/4 + Sum_{k>=1} (1-1/2^(4*k-2)) * (1-1/2)/(1-1/2^(4*k-1)) * f(2*k-1,4*k-1)/zeta(4*k-1) - (1-1/2^(4*k-3)) * f(4*k-2,4*k-2)/zeta(4*k-2) = 0.16205634516436945215..., where f(n,e) = Product_{prime p|n} (1-1/p)/(1-1/p^e).

			4 is a term since A381952(4) = gcd(4, A051903(4)) = gcd(4, 2) = 2.

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

Cf. A051903, A381952.

Subsequence of A368715 (numbers k such that A381952(k) >= 2).

q[k_] := GCD[k, Max[FactorInteger[k][[;;, 2]]]] == 2; Select[2*Range[200], q]

isok(k) = !(k % 2) && gcd(k, vecmax(factor(k)[, 2])) == 2;

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A368714 Numbers whose maximal exponent in their prime factorization is even.

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A381952 a(n) is the greatest common divisor of n and the maximum exponent in the prime factorization of n.

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A374587 The maximum exponent in the prime factorization of the numbers that are not coprime to the maximum exponent in their prime factorization.

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A381953 Numbers k such that A381952(k) = 2.

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