[seqfan] Re: The radical numbers

[Tomasz Ordowski]

P.S. SUPPLEMENT:

Cf. https://oeis.org/A027598 :
PRIME-PERFECT NUMBERS.

Regarding the semiprime phi-radical numbers:

[...] I don't know if there are infinitely many, or even any other
examples.  I do have a paper which discusses some similar problems,
such as rad(n) = rad(sigma(n)) and rad(n) = rad(phi(n)).  Maybe
some of the techniques of this paper can show there are infinitely
many phi-radicals or psi-radicals.  Here's a link to my paper:
https://math.dartmouth.edu/~carlp/pperfs13.pdf

Carl Pomerance [in a letter to me].

Perhaps now the topic will arouse interest.

I am asking for comments.

Thomas Ordowski

sob., 16 lut 2019 o 14:00 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> Dear SeqFans!
>
> The phi-radical numbers:
> Composite numbers n such that rad(phi(n)) = rad(n-1),
> where phi is the Euler totient function.
> 1729, 2431, 6601, 9605, 10585, 12801, 15211, 30889, 46657, 69751, 88561,
> 92929, 105001, 159895, 272323, 368641, 460801, 534061, 610051, 622909,
> 950797, 992251, ...
> These numbers are odd squarefree. They contain many Carmichael numbers.
> We only found the first such semiprime, namely 1525781251 = 19531 * 78121.
>
> The psi-radical numbers:
> Composite numbers n such that rad(psi(n)) = rad(n+1),
> where psi is the Dedekind function.
> 35, 161, 399, 899, 1349, 1457, 2015, 2915, 4199, 6479, 7055, 12995, 21869,
> 26751, 46079, 54755, 63503, 67199, 69695, 72029, 97019, 112499, 125315,
> 144399, 147455, 152099, 188441, 214199, 268583, 275561, 278963, 325247,
> 352835, 360149, 597617, 636803, 673595, 728999, 788543, 809999, 888719,
> 910115, ...
> They contain many Lucas-Carmichael numbers. However, we found here more
> semiprimes:
> 35, 161, 899, 1349, 1457, 21869, 278963, 360149, 728999,
> 27059269, 1106755649, ...
>
> Due to the symmetric definitions of the phi-radicals and the psi-radicals,
> we encourage to search for the second semiprime phi-radical numbers.
>
> Best regards,
>
> Amiram Eldar & Thomas Ordowski
> ______________________
> The sigma-radical numbers:
> Composite numbers n such that rad(sigma(n)) = rad(n+1),
> where sigma is the sum-of-divisors function.
> 35, 161, 399, 899, 1349, 1457, 2015, 2915, 2975, 4199, 6479, 7055, 12995,
> 21869, 26751, 46079, 54755, 63503, 67199, 69695, 72029, 97019, 112499,
> 122499, 125315, 144399, 147455, 152099, 188441, 214199, 219699, 268583,
> 275561, 278963, 325247, 352835, 360149, 597617, 614999, 636803, 673595,
> 728999, 788543, 809999, 888719, 896375, 910115, ...
> Note that the squarefree sigma-radical numbers are the psi-radical
> numbers.
> Those that are not squarefree: 2975, 122499, 219699, 614999, 896375, ...
>
>