cp's OEIS Frontend

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

%I A377020 #10 Oct 15 2024 09:49:31
%S A377020 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,
%T A377020 29,30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U A377020 55,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,73,74,75,76
%N A377020 Numbers whose prime factorization has exponents that are all numbers of the form m*k!, where 1 <= m <= k.
%C A377020 First differs from A138302 and A270428 at n = 57: a(57) = 64 is not a term of A138302 and A270428.
%C A377020 First differs from A337052 at n = 193: A337052(193) = 216 is not a term of this sequence.
%C A377020 First differs from A335275 at n = 227: A335275(227) = 256 is not a term of this sequence.
%C A377020 First differs from A220218 at n = 903: A220218(903) = 1024 is not a term of this sequence.
%C A377020 Numbers k such that A376886(k) = A001221(k).
%C A377020 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... .
%C A377020 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
%H A377020 Amiram Eldar, <a href="/A377020/b377020.txt">Table of n, a(n) for n = 1..10000</a>
%t A377020 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]
%o A377020 (PARI) isf(n) = {my(k = 2); while(!(n % k), n /= k; k++); n < k;}
%o A377020 is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isf(e[i]), return(0))); 1;}
%Y A377020 Cf. A001221, A051683, A376886, A377021, A377022.
%Y A377020 Subsequences: A005117, A004709, A377019.
%Y A377020 Cf. A138302, A220218, A270428, A335275, A337052.
%K A377020 nonn,easy
%O A377020 1,2
%A A377020 _Amiram Eldar_, Oct 13 2024