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

1,2

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.

Links

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Complement of A140831.

Cf. A005117.

Sequence in context: A353838 A183224 A357862 * A098240 A023805 A168186

Adjacent sequences: A383394 A383395 A383396 * A383398 A383399 A383400

Keyword

nonn,easy

Author

Boas Bakker, Apr 26 2025

Status

approved