A381952 -id:A381952 - cp's OEIS frontend

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

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;

A381954 The maximum exponent in the prime factorization of n that is coprime to n, or 0 if no such exponent exists.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 5, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Mar 11 2025

The sums of the first 10^k terms, for k = 1, 2, ..., are 11, 118, 1250, 12648, 126955, 1271426, 12720020, 127218134, 1272236949, 12722547575, .... . Apparently, the asymptotic mean of this sequence equals 1.2722... .

			a(2) = 1 since 2 = 2^1 and 1 is coprime to 2.
a(4) = 0 since 4 = 2^2 and the exponent 2 is not coprime to 4.
a(12) = 1 since 12 = 2^2 * 3^1, the exponent 2 is not coprime to 12, and the exponent 1 is coprime to 12.

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

Cf. A051903, A381952.

a[n_] := If[(s = Select[FactorInteger[n][[;; , 2]], CoprimeQ[#, n] &]) != {}, Max[s], 0]; a[1] = 0; Array[a, 100]

a(n) = {my(e = factor(n)[, 2], s = select(x -> if(gcd(x, n) == 1, x, 0), e)); if(#s == 0, 0, vecmax(s));}

cp's OEIS Frontend

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

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