A360793 -id:A360793 - cp's OEIS frontend

A375076 Numbers whose prime factorization exponents include at least one 1, at least one 3 and no other exponents.

24, 40, 54, 56, 88, 104, 120, 135, 136, 152, 168, 184, 189, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 344, 351, 375, 376, 378, 408, 424, 440, 456, 459, 472, 488, 513, 520, 536, 552, 568, 584, 594, 616, 621, 632, 664, 680, 686, 696, 702, 712, 728, 744, 750

Offset: 1

Amiram Eldar, Jul 29 2024

First differs from its subsequence A360793 at n = 79: a(79) = 1080 = 2^3 * 3^3 * 5 is not a term of A360793.
Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {1, 3}.
The asymptotic density of this sequence is ((zeta(6)/zeta(3)) * Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5) - 1)/zeta(2) = 0.076359822332835689478... .

Equals A336591 \ (A005117 UNION A062838).
Subsequences: A065036, A360793.
Cf. A002117, A013661, A013664, A136568.

Select[Range[750], Union[FactorInteger[#][[;; , 2]]] == {1, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [1, 3];
```

A375934 Numbers whose prime factorization has a second-largest exponent that equals 1.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 204

Offset: 1

Amiram Eldar, Sep 03 2024

First differs from A332785 at n = 112: A332785(112) = 360 = 2^3 * 3^2 * 5 is not a term of this sequence.
First differs from A317616 at n = 38: A317616(38) = 144 = 2*4 * 3^2 is not a term of this sequence.
Numbers k such that A375933(k) = 1.
Numbers of the form s1 * s2^e, where s1 and s2 are coprime squarefree numbers that are both larger than 1, and e >= 2.
The asymptotic density of this sequence is Sum_{e>=2} d(e) = 0.36113984820338109927..., where d(e) = Product_{p prime} (1 - 1/p^2 + 1/p^e - 1/p^(e+1)) - Product_{p prime} (1 - 1/p^(e+1)) is the asymptotic density of terms k with A051903(k) = e >= 2.

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

Equals A375142 \ A072777.
Subsequence of A013929.
Subsequences: A054753, A065036, A072357, A095990, A096156, A178739, A178740, A179664, A179668, A179692, A189987, A360793, A375076.
Cf. A001221, A005117, A051903, A051904, A072777, A317616, A332785, A375142, A375933.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[0, Max[Select[e, # < Max[e] &]]] == 1]; Select[Range[300], q]

is(n) = if(n == 1, 0, my(e = factor(n)[,2]); e = select(x -> x < vecmax(e), e); if(#e == 0, 0, vecmax(e) == 1));

A051904(a(n)) = 1.
A051903(a(n)) >= 2.
A001221(a(n)) = 2.

cp's OEIS Frontend

A375076 Numbers whose prime factorization exponents include at least one 1, at least one 3 and no other exponents.

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