A377019 Numbers whose prime factorization has exponents that are all factorial numbers.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78

Offset

1,2

Comments

First differs from its subsequence A004709 and from A344742 at n = 55: a(55) = 64 = 2^6 is not a term of A004709 and A344742.

Numbers k such that A376885(k) = A001221(k).

The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + (1 - 1/p) * (Sum_{k>=3} 1/p^(k!))) = 0.84018238588352905855... .

Links

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

Mathematica

factorialQ[n_] := factorialQ[n] = Module[{m = n, k = 2}, While[Divisible[m, k], m /= k; k++]; m == 1]; q[n_] := AllTrue[FactorInteger[n][[;; , 2]], factorialQ]; Select[Range[100], q]

Prog

(PARI) isf(n) = {my(k = 2); while(!(n % k), n /= k; k++); n == 1; }
is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isf(e[i]), return(0))); 1; }

Crossrefs

Cf. A000142, A001221, A344742, A376885, A377021, A377022.

Subsequence of A377020.

Subsequences: A005117, A004709.

Sequence in context: A043093 A023802 A007915 * A344742 A004709 A048107

Adjacent sequences: A377016 A377017 A377018 * A377020 A377021 A377022

Keyword

nonn,easy

Author

Amiram Eldar, Oct 13 2024

Status

approved