A360723 Numbers that have at least one exponent in their canonical prime factorization that is neither 2 nor of the form 2^k-1, k>=1.

16, 32, 48, 64, 80, 81, 96, 112, 144, 160, 162, 176, 192, 208, 224, 240, 243, 256, 272, 288, 304, 320, 324, 336, 352, 368, 400, 405, 416, 432, 448, 464, 480, 486, 496, 512, 528, 544, 560, 567, 576, 592, 608, 624, 625, 648, 656, 672, 688, 704, 720, 729, 736, 752

Offset: 1

Author

Amiram Eldar, Feb 18 2023

Keywords

• nonn
• easy

Comments

Numbers that have at least one powerful divisor that is not infinitary divisor, i.e., numbers k such that A360721(k) < A005361(k).

The complement of this sequence is the sequence of numbers all of whose powerful divisors are also infinitary divisors. The related sequence of numbers all of whose infinitary divisors are powerful is the sequence of squares (A000290).

The asymptotic density of this sequence is 1 - Product_{p prime} ((1 - 1/p) * (1 + 1/p^2 + Sum_{i>=1} 1/p^(2^i-1))) = 0.071899867098952952524... .

Links

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

Crossrefs

Cf. A000225, A000290, A001694, A005361, A077609, A360721.

Programs

Mathematica

q[n_] := AnyTrue[FactorInteger[n][[;; , 2]], # != 2 && # + 1 != 2^IntegerExponent[# + 1, 2] &]; Select[Range[1000], q]

PARI

is(n) = {my(e = factor(n)[, 2]); for(i = 1, #e, if(e[i] != 2 && (e[i]+1)>>valuation(e[i]+1, 2) != 1, return(1))); 0;}