A335989 Terms of A301517 that are not exponentially odd numbers (A268335).

12500, 18252, 21600, 37500, 50000, 67228, 84500, 87500, 91260, 127764, 137500, 146016, 150000, 151200, 162500, 200000, 200772, 201684, 212500, 231868, 237500, 237600, 253500, 262500, 268912, 274400, 280800, 287500, 310284, 336140, 337500, 346788, 350000, 362500

Offset: 1

Author

Amiram Eldar, Jul 03 2020

Keywords

• nonn

Comments

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.

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.

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).

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A048250, A059956, A065463, A162296, A268335, A301517.

Programs

Mathematica

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] &]