A392631 - OEIS

%I #6 Jan 18 2026 15:32:06

%S 1,2,3,5,6,7,8,10,11,13,14,15,17,19,21,22,23,24,26,27,29,30,31,33,34,

%T 35,37,38,39,40,41,42,43,46,47,51,53,54,55,56,57,58,59,61,62,65,66,67,

%U 69,70,71,73,74,77,78,79,82,83,85,86,87,88,89,91,93,94,95,97

%N Numbers whose exponential divisors are all numbers whose number of divisors is a power of 2.

%C Subsequence of A036537 and first differs from it at n = 22546: A036537(22546) = 32768 = 2^15 is not a term in this sequence.

%C First differs from its subsequence A336591 at n = 89: a(89) = 128 is not a term in A336591.

%C Numbers k such that A392630(k) = A049419(k).

%C Numbers whose set of distinct prime factorization exponents is a subset of {1} U A000668.

%C The asymptotic density of this sequence is Product_{p prime} (1-1/p) * (1 + 1/p + Sum_{k>=1} 1/p^A000668(k)) = 0.68781424907634032654... .

%H Amiram Eldar, <a href="/A392631/b392631.txt">Table of n, a(n) for n = 1..10000</a>

%t q[k_] := AllTrue[FactorInteger[k][[;;, 2]], # == 1 || (PrimeQ[#] && #+1 == 2^IntegerExponent[#+1, 2]) &]; Select[Range[100], q]

%o (PARI) isexp(e) = e == 1 || (isprime(e) && e+1 == 1 << valuation(e + 1, 2));

%o isok(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isexp(e[i]), return(0))); 1;}

%Y Subsequence of A036537.

%Y A336591 is a subsequence.

%Y Cf. A000668, A049419, A322791, A392630, A392632.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Jan 18 2026