A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 204, 207, 208, 212, 220, 228, 234, 236

Offset: 1

Author

Amiram Eldar, Feb 19 2025

Keywords

• nonn
• easy

Comments

Numbers k whose largest unitary divisor that is a square, A350388(k), is a prime power (A246655), or equivalently, A350388(k) is in A056798 \ {1}.

Numbers having exactly one non-unitary prime factor and its multiplicity is even.

Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m} with m >= 1, i.e., any number (including zero) of 1's and then a single even number.

The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} p/((p-1)*(p+1)^2) = 0.24200684327095676029... .

Links

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

Crossrefs

Complement of A381312 within A190641.

Cf. A057521, A118914, A246655, A350388.

Subsequence of: A377816.

Subsequences: A001248, A030514, A030516, A030629, A030631, A054753, A056798 \ {1}, A085987, A178739, A179644, A179645, A179668, A179672, A179693, A179747, A189982, A189983, A189985, A189987, A190110, A190292, A190388.

Programs

Mathematica

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;;,2]]]}, EvenQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000],q]

PARI

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); !(e[1] % 2) && (#e == 1 || e[2] == 1));