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

Author

Amiram Eldar, Oct 13 2024

Keywords

• nonn
• easy

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

Crossrefs

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

Subsequence of A377020.

Subsequences: A005117, A004709.

Programs

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]

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;}