A140831 -id:A140831 - cp's OEIS frontend

A379094 Numbers whose factors in the canonical prime factorization neither increase weakly nor decrease weakly.

60, 84, 90, 120, 126, 132, 156, 168, 180, 204, 228, 240, 252, 264, 270, 276, 280, 300, 312, 315, 336, 348, 350, 360, 372, 378, 408, 420, 440, 444, 456, 480, 492, 495, 504, 516, 520, 525, 528, 540, 550, 552, 560, 564, 585, 588, 594, 600, 616, 624, 630, 636, 650

Offset: 1

Peter Luschny, Dec 17 2024

nonn

A379097 is a subsequence.
From Michael De Vlieger, Dec 18 2024: (Start)
Proper subset of A126706.
Smallest powerful number is a(314) = 2700. (End)

			60 is a term because the factors in the canonical prime factorization are [4, 3, 5], a list that is neither increasing nor decreasing.
Primorials (A002110) are not terms of this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A053585, A057714, A085232, A140831, A379097.

with(ArrayTools):
fact := n -> local p; [seq(p[1]^p[2], p in ifactors(n)[2])]:
isA379094 := proc(n) local f; f := fact(n);
is(not IsMonotonic(f, direction=decreasing, strict=false) and not IsMonotonic(f, direction=increasing, strict=false)) end:
select(isA379094, [seq(1..650)]);

Select[Range[650], Function[f, NoneTrue[{Sort[f], ReverseSort[f]}, # == f &]][Power @@@ FactorInteger[#]] &] (* Michael De Vlieger, Dec 18 2024 *)

is_a379094(n) = my(C=apply(x->x[1]^x[2], Vec(factor(n)~))); vecsort(C)!=C && vecsort(C,,4)!=C \\ Hugo Pfoertner, Dec 18 2024

A383397 Numbers in whose canonical prime factorization the powers of the primes form a strictly increasing sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101

Offset: 1

Boas Bakker, Apr 26 2025

Alternative name: positive integers with canonical prime factorization p_1 ^ e_1 * p_2 ^ e_2 * ... * p_k ^ e_k which satisfy p_1 ^ e_1 < p_2 ^ e_2 < ... < p_k ^ e_k.

The asymptotic density of this sequence seems to be about 0.84.

			18 = 2^1 * 3^2 is in the sequence as 2^1 < 3^2.
12 is not in the sequence because 12 = 2^2 * 3^1 and 4>3.

Boas Bakker, Table of n, a(n) for n = 1..100000

Complement of A140831.

Cf. A005117.

Select[Range[100], Less @@ Power @@@ FactorInteger[#] &] (* Amiram Eldar, Apr 26 2025 *)

is(n) = {my(f = factor(n), r = 0); for(i = 1, #f~, c = f[i,1]^f[i,2]; if(c > r, r = c, return(0))); 1} \\ David A. Corneth, Apr 26 2025

cp's OEIS Frontend

A379094 Numbers whose factors in the canonical prime factorization neither increase weakly nor decrease weakly.

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