A374588 -id:A374588 - cp's OEIS frontend

A374590 Numbers whose maximum exponent in their prime factorization is an evil number (A001969).

8, 24, 27, 32, 40, 54, 56, 64, 72, 88, 96, 104, 108, 120, 125, 135, 136, 152, 160, 168, 184, 189, 192, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 320, 328, 343, 344, 351, 352, 360, 375, 376, 378, 392, 408, 416, 424, 440, 448, 456, 459

Offset: 1

Amiram Eldar, Jul 12 2024

The asymptotic density of this sequence is Sum_{k in A001969} (1/zeta(k+1) - 1/zeta(k)) = 0.12101890210392912747... .

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

Cf. A001969, A051903.
Subsequence of A013929 and A262675 \ {1}.
Similar sequences: A368714, A369937, A369938, A369939, A374588, A374589.

evilQ[n_] := EvenQ[DigitCount[n, 2, 1]]; q[n_] := evilQ[Max[FactorInteger[n][[;; , 2]]]]; Select[Range[500], q]

is(n) = n > 1 && !(hammingweight(vecmax(factor(n)[, 2])) % 2);

A374589 Numbers whose maximum exponent in their prime factorization is a powerful number larger than 1.

Original entry on oeis.org

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 256, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 768, 784, 810, 816, 848, 880, 891, 912, 944, 976, 1008, 1040, 1053, 1072, 1104, 1134, 1136, 1168, 1200

Offset: 1

Amiram Eldar, Jul 12 2024

Subsequence of A130897 and first differs from it at n = 115: A130897(115) = 2592 = 2^5 * 3^4 is not a term of this sequence.

The asymptotic density of this sequence is d = Sum_{k > 1 and in A001694} (1/zeta(k+1) - 1/zeta(k)) = 0.043523813088759413253... . The asymptotic density of this sequence within A130897 is d/(1 - A262276) = 0.98744988886705430331... .

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

Cf. A001694, A051903, A262276, A361177.
Subsequence of A013929, A130897 and A372405.
Similar sequences: A368714, A369937, A369938, A369939, A374588, A374590.

powQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 1; q[n_] := powQ[Max[ FactorInteger[n][[;; , 2]] ]]; Select[Range[1200], q]

ispow(n) = n > 1 && ispowerful(n);
is(n) = n > 1 && ispow(vecmax(factor(n)[, 2]))

cp's OEIS Frontend

A374590 Numbers whose maximum exponent in their prime factorization is an evil number (A001969).

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