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

%I A377022 #9 Oct 13 2024 11:18:16
%S A377022 1,16,81,625,1296,2401,4096,10000,14641,28561,38416,50625,65536,83521,
%T A377022 130321,194481,234256,262144,279841,331776,456976,531441,707281,
%U A377022 810000,923521,1185921,1336336,1500625,1874161,2085136,2313441,2560000,2825761,3111696,3418801
%N A377022 Numbers whose prime factorization has exponents that have no digit 1 in their factorial-base representation (A255411).
%C A377022 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.
%H A377022 Amiram Eldar, <a href="/A377022/b377022.txt">Table of n, a(n) for n = 1..10000</a>
%F A377022 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^A255411(k)) = 1.07819745085315583226... .
%t A377022 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]
%o A377022 (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;}
%o A377022 is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isexp(e[i]), return(0))); 1;}
%Y A377022 Cf. A255411, A376885, A377019, A377020, A377021.
%Y A377022 Analogous to A001694.
%Y A377022 Subsequence of A036967.
%K A377022 nonn,base
%O A377022 1,2
%A A377022 _Amiram Eldar_, Oct 13 2024