

A335989


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


2



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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 are not exceeding 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).


LINKS

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


EXAMPLE

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.


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] &]


CROSSREFS

Cf. A048250, A059956, A065463, A162296, A268335, A301517.
Sequence in context: A072891 A003416 A178278 * A220026 A349044 A231956
Adjacent sequences: A335986 A335987 A335988 * A335990 A335991 A335992


KEYWORD

nonn


AUTHOR

Amiram Eldar, Jul 03 2020


STATUS

approved



