A381950 -id:A381950 - cp's OEIS frontend

A381956 Numbers k such that k and the maximum exponent in the prime factorization of k have opposite parities.

1, 2, 6, 8, 9, 10, 14, 22, 24, 25, 26, 30, 32, 34, 38, 40, 42, 45, 46, 49, 54, 56, 58, 62, 63, 66, 70, 72, 74, 75, 78, 81, 82, 86, 88, 94, 96, 99, 102, 104, 106, 108, 110, 114, 117, 118, 120, 121, 122, 128, 130, 134, 136, 138, 142, 146, 147, 152, 153, 154, 158, 160

Offset: 1

Amiram Eldar, Mar 11 2025

Numbers k such that A000035(k) != A381955(k).

The asymptotic density of this sequence is 1/2 + Sum_{k>=2} ((-1)^(k+1)/((2^k-1)*zeta(k))) = 0.3753176676637566708086... .

			2 is a term since 2 mod 2 = 0 and A051903(2) mod 2 = 1 mod 2 = 1 != 0.
4 is not a term since 4 mod 2 = 0 and also A051903(4) mod 2 = 2 mod 2 = 0.

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

Cf. A000035, A051903, A381955.

Subsequences: A039956, A381950.

q[k_] := If[k == 1, True, Mod[k, 2] != Mod[Max[FactorInteger[k][[;; , 2]]], 2]]; Select[Range[160], q]

isok(k) = if(k == 1, 1, k % 2 != vecmax(factor(k)[,2]) % 2);

A381951 Nonsquarefree odd numbers whose prime factorization has an odd maximum exponent.

Original entry on oeis.org

27, 125, 135, 189, 243, 297, 343, 351, 375, 459, 513, 621, 675, 783, 837, 875, 945, 999, 1029, 1107, 1125, 1161, 1215, 1269, 1323, 1331, 1375, 1431, 1485, 1593, 1625, 1647, 1701, 1715, 1755, 1809, 1917, 1971, 2079, 2125, 2133, 2187, 2197, 2241, 2295, 2375, 2403, 2457

Offset: 1

Amiram Eldar, Mar 11 2025

Nonsquarefree odd numbers k such that A051903(k) is odd, or equivalently, odd numbers k such that A051903(k) is an odd number that is larger than 1.

The asymptotic density of this sequence is (1/2) * Sum_{k>=3} (-1)^(k+1) * (1 - 2^k/((2^k-1)*zeta(k))) = 0.019098071299657074975... .

			125 = 5^3 is a term since it is odd, divisible by a square (25, and thus it is nonsquarefree), and the maximum exponent in its prime factorization is 3, which is odd.

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

Intersection of A005408 and A376142.
Complement of A381950 within A013929.
Cf. A051903.

q[n_] := n > 1 && OddQ[n]; Select[Range[1, 2500, 2], q[Max[FactorInteger[#][[;; , 2]]]] &]

isok(k) = k > 1 && k % 2 && apply(x -> (x > 1 && x % 2), vecmax(factor(k)[, 2]));

cp's OEIS Frontend

A381956 Numbers k such that k and the maximum exponent in the prime factorization of k have opposite parities.

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