A306479 Squarefree composite numbers m such that rad(p-1) = rad(m-1) for every prime p dividing m.

1729, 46657, 1525781251

Offset

1,1

Comments

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, ...

The following semiprimes also belong to this sequence: 763546828801, 6031047559681, 184597450297471, 732785991945841, 18641350656000001, 55212580317094201. - Daniel Suteu, Feb 18 2019

Links

Table of n, a(n) for n=1..3.

Carlos Rivera, Puzzle 969. Rad(m - 1) = Rad(phi(m)), The Prime Puzzles & Problems Connection.

Prog

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

Crossrefs

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

Sequence in context: A284671 A265328 A265628 * A272798 A212920 A317126

Adjacent sequences: A306476 A306477 A306478 * A306480 A306481 A306482

Keyword

nonn,more

Author

Amiram Eldar and Thomas Ordowski, Feb 18 2019

Status

approved