login
Squarefree composite numbers m such that rad(p-1) = rad(m-1) for every prime p dividing m.
1

%I #41 Sep 28 2019 08:39:57

%S 1729,46657,1525781251

%N Squarefree composite numbers m such that rad(p-1) = rad(m-1) for every prime p dividing m.

%C a(1) and a(2) are Carmichael numbers (no more such Carmichael numbers up to 10^18), a(3) = (5^7-4)*(5^7-1)/4 is semiprime. The semiprimes of the form (b^p - (b-1))*(b^p - 1)/(b-1) of this sequence include (3^541-2)*(3^541-1)/2, (5^7-4)*(5^7-1)/4, (5^47-4)*(5^47-1)/4, (17^11-16)*(17^11-1)/16, (65^19-64)*(65^19-1)/64, (129^5-128)*(129^5-1)/128, ...

%C The following semiprimes also belong to this sequence: 763546828801, 6031047559681, 184597450297471, 732785991945841, 18641350656000001, 55212580317094201. - _Daniel Suteu_, Feb 18 2019

%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.

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

%o isok(m) = {if ((m==1) || isprime(m) || !issquarefree(m), return(0)); my(f = factor(m)[,1], r = rad(m-1)); for (i=1, #f, if (rad(f[i]-1) != r, return (0));); return (1);} \\ _Michel Marcus_, Feb 18 2019

%Y Cf. A002997, A007947, A306478 (phi-radical numbers).

%K nonn,more

%O 1,1

%A _Amiram Eldar_ and _Thomas Ordowski_, Feb 18 2019