A377020 Numbers whose prime factorization has exponents that are all numbers of the form m*k!, where 1 <= m <= k.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 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, 48, 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

Offset: 1

Author

Amiram Eldar, Oct 13 2024

Keywords

• nonn
• easy

Comments

First differs from A138302 and A270428 at n = 57: a(57) = 64 is not a term of A138302 and A270428.

First differs from A337052 at n = 193: A337052(193) = 216 is not a term of this sequence.

First differs from A335275 at n = 227: A335275(227) = 256 is not a term of this sequence.

First differs from A220218 at n = 903: A220218(903) = 1024 is not a term of this sequence.

Numbers k such that A376886(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^A051683(k))) = 0.87902453718626485582... .

a(n) = A096432(n-1) for 2<=n<380, but then the sequences start to differ: A096432 contains 432, 648, 1024, 1728, 2000, 2160,... which are not in this sequence. - R. J. Mathar, Oct 15 2024

Links

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

Crossrefs

Cf. A001221, A051683, A376886, A377021, A377022.

Subsequences: A005117, A004709, A377019.

Cf. A138302, A220218, A270428, A335275, A337052.

Programs

Mathematica

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

PARI

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