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 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 (list; graph; refs; listen; history; text; internal format)
 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 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..10000 StackExchange, Question 73354 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), 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), 1, 0), i=ifactors(k)) do od; k     end: seq(a(n), n=1..80);  # Alois P. Heinz, Jun 30 2016 MATHEMATICA Select[Range, Union[IntegerQ /@ Sqrt[Transpose[FactorInteger[#]][]]][] &] (* 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

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Last modified April 22 12:53 EDT 2021. Contains 343177 sequences. (Running on oeis4.)