

A360723


Numbers that have at least one exponent in their canonical prime factorization that is neither 2 nor of the form 2^k1, 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
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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^i1))) = 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



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



