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%I #18 Feb 17 2024 11:27:38
%S 12500,18252,21600,37500,50000,67228,84500,87500,91260,127764,137500,
%T 146016,150000,151200,162500,200000,200772,201684,212500,231868,
%U 237500,237600,253500,262500,268912,274400,280800,287500,310284,336140,337500,346788,350000,362500
%N Terms of A301517 that are not exponentially odd numbers (A268335).
%C If k = Product p^e, then A162296(k) / A048250(k) = -1 + Product (p^(e+1) - 1)/(p^2 - 1). If k is exponentially odd, then e = 2*m - 1 is odd for all the prime factors p of k and p^(e+1) - 1 = (p^2)^m - 1 is divisible by p^2 - 1. Therefore, A162296(k) / A048250(k) is an integer for all exponentially odd numbers, and it is a positive integer for all the nonsquarefree (A013929) exponentially odd numbers.
%C It seems that most of the terms of A301517 are exponentially odd numbers. For example, the first 10^4 terms of A301517 include only 9 terms that are not exponentially odd numbers. Up to 10^8 there are 9660732 terms of A301517, and only 9107 of them are not exponentially odd numbers.
%C The number of terms of this sequence that do not exceed 10^k, for k = 5, 6, ... are 9, 92, 916, 9107, 91172, 911187, .... Apparently, this sequence has an asymptotic density c = 0.000091... If this is true, then the asymptotic density of A301517 is c + A065463 - A059956 = 0.096606... (A065463 is the density of the exponentially odd numbers, and A059956 is the density of the squarefree numbers which are a subset of the exponentially odd numbers).
%H Amiram Eldar, <a href="/A335989/b335989.txt">Table of n, a(n) for n = 1..10000</a>
%e 12500 = 2^2 * 5^5 is a term since the exponent of its prime factor 2 is 2 which even, and therefore it is not an exponentially odd number, and the sum of its squarefree divisors, A048250(12500) = 18 divides the sum of its nonsquarefree divisors, A162296(12500) = 27324 = 18 * 1518.
%t f[p_, e_] := (p^(e + 1) - 1)/(p^2 - 1); Select[Range[2, 4*10^5], Max[Last /@ (fct = FactorInteger[#])] > 1 && ! AllTrue[Last /@ fct, OddQ] && (r = Times @@ (f @@@ fct)) > 1 && IntegerQ[r] &]
%Y Cf. A048250, A059956, A065463, A162296, A268335, A301517.
%K nonn
%O 1,1
%A _Amiram Eldar_, Jul 03 2020
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