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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
MATHEMATICA
q[n_] := AnyTrue[FactorInteger[n][[;; , 2]], # != 2 && # + 1 != 2^IntegerExponent[# + 1, 2] &]; Select[Range[1000], q]
PROG
(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; }
CROSSREFS
Sequence in context: A048111 A122614 A046101 * A044856 A217558 A044901
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Feb 18 2023
STATUS
approved

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Last modified May 25 12:29 EDT 2024. Contains 372788 sequences. (Running on oeis4.)