A197680 Numbers whose exponents in their prime power factorization are squares.

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

Author

A. Neves, Oct 17 2011

Keywords

• nonn

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 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
• Math 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.

Crossrefs

Cf. A000290, A049419, A005117, A115063, A209061.

Programs

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 *)

PARI

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

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

Extensions

Reformulation of the name by Vladimir Shevelev, Oct 14 2015