

A197680


Numbers whose exponents in their prime power factorization are squares.


7



1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101
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OFFSET

1,2


COMMENTS

Numbers whose prime factorization has the form Product_i p_i^e_i where the e_i are all squares.
All squarefree numbers (A005117) are in the sequence.  Vladimir Shevelev, Nov 16 2015
Let h_k be the density of the subsequence of A197680 of numbers whose prime power factorization (PPF) has the form Product_i p_i^e_i where the e_i all squares <= k^2. Then for every k>1 there exists eps_k>0 such that for any x from the interval (h_keps_k, h_k) there is no a sequence S of positive integers such that x is the density of numbers whose PPF has the form Product_i p_i^e_i where the e_i are all in S.  For a proof, see [Shevelev], second link.  Vladimir Shevelev, Nov 17 2015


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000
StackExchange, Question 73354
Vladimir Shevelev, Exponentially Snumbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially Snumbers, arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, Sexponential numbers, Acta Arithmetica, Vol. 175(2016), 385395.


FORMULA

Sum_{i<=x, i is in A197680} 1 = h*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c=4*sqrt(2.4/log 2)=7.44308... and h=Product_{prime p} (1+Sum_{i>=2} (u(i)u(i1))/p^i)=0.641115... where u(n) is the characteristic function of sequence A000290. The calculations of h in the formula were done independently by Juan AriasdeReyna and Peter J. C. Moses. For a proof of the formula, see the first Shevelev link.  Vladimir Shevelev, Nov 17 2015


MAPLE

a:= proc(n) option remember; local k; for k from 1+
`if`(n=1, 0, a(n1)) while 0=mul(`if`(issqr(
i[2]), 1, 0), i=ifactors(k)[2]) do od; k
end:
seq(a(n), n=1..80); # Alois P. Heinz, Jun 30 2016


MATHEMATICA

Select[Range[100], Union[IntegerQ /@ Sqrt[Transpose[FactorInteger[#]][[2]]]][[1]] &] (* T. D. Noe, Oct 18 2011 *)


PROG

(PARI) isok(n) = {my(f = factor(n)[, 2]); #select(x>issquare(x), f) == #f; } \\ Michel Marcus, Oct 23 2015


CROSSREFS

Cf. A000290, A005117, A115063, A197680, A209061.
Sequence in context: A252895 A336224 A274034 * A119024 A321453 A203076
Adjacent sequences: A197677 A197678 A197679 * A197681 A197682 A197683


KEYWORD

nonn


AUTHOR

A. Neves, Oct 17 2011


EXTENSIONS

Reformulation of the name by Vladimir Shevelev, Oct 14 2015


STATUS

approved



