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A197680
Numbers whose exponents in their prime power factorization are squares.
18
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
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_k-eps_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
Numbers with an odd number of exponential divisors (A049419). - Amiram Eldar, Nov 05 2021
LINKS
StackExchange, Question 73354, 2011.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.
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(i-1))/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 Arias-de-Reyna 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(n-1)) 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
KEYWORD
nonn
AUTHOR
A. Neves, Oct 17 2011
EXTENSIONS
Reformulation of the name by Vladimir Shevelev, Oct 14 2015
STATUS
approved