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

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

Author

Boas Bakker, Apr 26 2025

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

Complement of A140831.

Cf. A005117.

Programs

Mathematica

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

PARI

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