[seqfan] The radical numbers

[Tomasz Ordowski]

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, ...