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

1729, 2431, 6601, 9605, 10585, 12801, 15211, 30889, 46657, 69751, 88561, 92929, 105001, 159895, 272323, 368641, 460801, 534061, 610051, 622909, 950797, 992251, 1047619, 1372801, 1374895, 1745701, 1902691, 2210671, 2628073, 2704801, 3225601, 5629339, 5690251, 6840001, 9738751

Offset

1,1

Comments

Problem: are there infinitely many such numbers?

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

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

Links

Table of n, a(n) for n=1..35.

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

Mathematica

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

Prog

(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
isok(m) = !isprime(m) && (rad(eulerphi(m)) == rad(m-1)); \\ Michel Marcus, Feb 18 2019

Crossrefs

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

Sequence in context: A044883 A288153 A154717 * A051388 A033181 A300949

Adjacent sequences: A306475 A306476 A306477 * A306479 A306480 A306481

Keyword

nonn

Author

Amiram Eldar and Thomas Ordowski, Feb 18 2019

Status

approved