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
Alois P. Heinz, Table of n, a(n) for n = 1..10000
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