cp's OEIS Frontend

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.

%I A360723 #24 May 27 2023 05:18:51
%S A360723 16,32,48,64,80,81,96,112,144,160,162,176,192,208,224,240,243,256,272,
%T A360723 288,304,320,324,336,352,368,400,405,416,432,448,464,480,486,496,512,
%U A360723 528,544,560,567,576,592,608,624,625,648,656,672,688,704,720,729,736,752
%N 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.
%C A360723 Numbers that have at least one powerful divisor that is not infinitary divisor, i.e., numbers k such that A360721(k) < A005361(k).
%C A360723 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).
%C A360723 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... .
%H A360723 Amiram Eldar, <a href="/A360723/b360723.txt">Table of n, a(n) for n = 1..10000</a>
%t A360723 q[n_] := AnyTrue[FactorInteger[n][[;; , 2]], # != 2 && # + 1 != 2^IntegerExponent[# + 1, 2] &]; Select[Range[1000], q]
%o A360723 (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;}
%Y A360723 Cf. A000225, A000290, A001694, A005361, A077609, A360721.
%K A360723 nonn,easy
%O A360723 1,1
%A A360723 _Amiram Eldar_, Feb 18 2023