A377022 Numbers whose prime factorization has exponents that have no digit 1 in their factorial-base representation (A255411).

1, 16, 81, 625, 1296, 2401, 4096, 10000, 14641, 28561, 38416, 50625, 65536, 83521, 130321, 194481, 234256, 262144, 279841, 331776, 456976, 531441, 707281, 810000, 923521, 1185921, 1336336, 1500625, 1874161, 2085136, 2313441, 2560000, 2825761, 3111696, 3418801

Offset: 1

Author

Amiram Eldar, Oct 13 2024

Keywords

• nonn
• base

Comments

Numbers that are "powerful" when they are factorized into factors of the form p^(k!), where p is a prime and k >= 1, a factorization that is done using the factorial-base representation of the exponents in the prime factorization (see A376885 for more details). Each factor p^(k!) has a multiplicity that is larger than 1.

Links

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

Crossrefs

Cf. A255411, A376885, A377019, A377020, A377021.

Analogous to A001694.

Subsequence of A036967.

Programs

Mathematica

expQ[n_] := expQ[n] = Module[{k = n, m = 2, r, s = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r == 1, s = 0; Break[]]; m++]; s == 1]; seq[lim_] := Module[{p = 2, s = {1}, emax, es}, While[(emax = Floor[Log[p, lim]]) > 3, es = Select[Range[0, emax], expQ]; s = Union[s, Select[Union[Flatten[Outer[Times, s, p^es]]], # <= lim &]]; p = NextPrime[p]]; s]; seq[4*10^6]

PARI

isexp(n) = {my(k = n, m = 2, r); while([k, r] = divrem(k, m); k != 0 || r != 0, if(r == 1, return(0)); m++); 1;}
is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isexp(e[i]), return(0))); 1;}

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^A255411(k)) = 1.07819745085315583226... .