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A306478 The phi-radical numbers: composite numbers m such that rad(phi(m)) = rad(m-1). 1

%I #36 Jan 10 2020 05:27:38

%S 1729,2431,6601,9605,10585,12801,15211,30889,46657,69751,88561,92929,

%T 105001,159895,272323,368641,460801,534061,610051,622909,950797,

%U 992251,1047619,1372801,1374895,1745701,1902691,2210671,2628073,2704801,3225601,5629339,5690251,6840001,9738751

%N The phi-radical numbers: composite numbers m such that rad(phi(m)) = rad(m-1).

%C Problem: are there infinitely many such numbers?

%C These numbers are odd squarefree. They contain many Carmichael numbers.

%C The smallest such semiprime is 1525781251 = 19531*78121, see A306479.

%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_969.htm">Puzzle 969. Rad(m - 1) = Rad(phi(m))</a>, The Prime Puzzles & Problems Connection.

%t rad[n_] := Times @@ (First@# & /@ FactorInteger@n); Select[Range[100000], CompositeQ[#] && rad[EulerPhi[#]] == rad[# - 1] &]

%o (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947

%o isok(m) = !isprime(m) && (rad(eulerphi(m)) == rad(m-1)); \\ _Michel Marcus_, Feb 18 2019

%Y Cf. A000010, A002997, A007947, A055744 (rad(phi(n)) = rad(n)), A306479.

%K nonn

%O 1,1

%A _Amiram Eldar_ and _Thomas Ordowski_, Feb 18 2019

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)