A392631 Numbers whose exponential divisors are all numbers whose number of divisors is a power of 2.

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, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97

Offset

1,2

Comments

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

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

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

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

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... .

Links

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

Mathematica

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

Prog

(PARI) isexp(e) = e == 1 || (isprime(e) && e+1 == 1 << valuation(e + 1, 2));
isok(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isexp(e[i]), return(0))); 1; }

Crossrefs

Subsequence of A036537.

A336591 is a subsequence.

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

Sequence in context: A268335 A002035 A336591 * A036537 A072510 A084116

Adjacent sequences: A392628 A392629 A392630 * A392632 A392633 A392634

Keyword

nonn,easy

Author

Amiram Eldar, Jan 18 2026

Status

approved