A180248 Odd composite squarefree numbers k such that r = 2*(p - 2 + k/p)/(p-1) is an integer for each prime divisor p of k.

15, 91, 435, 561, 703, 1105, 1729, 1891, 2465, 2701, 2821, 3367, 5551, 6601, 8695, 8911, 10585, 11305, 12403, 13981, 15051, 15841, 16471, 18721, 23001, 26335, 29341, 30889, 38503, 39865, 41041, 46657, 49141, 52633, 53131, 62745

Offset

1,1

Comments

Conjecture: k is a Carmichael number (A002997) if and only if k is a term of this sequence and all r-values of k are even.

From Ridouane Oudra, Apr 28 2019: (Start)

This sequence can also be defined as: Odd composite squarefree numbers k such that r' = 2*(k-1)/(p-1) is an integer for each prime divisor p of k. Proof:

2*(p - 2 + k/p)/(p-1) + 2*(k/p-1) = 2*(k-1)/(p-1),

so r is an integer if and only if r' is. (2*(k/p-1) is always an integer.)

With this new definition and Korselt's theorem it is easily shown that the proposed conjecture is true.

(End)

Links

K. Brockhaus, Table of n, a(n) for n = 1..653 (terms < 10^8)

Prog

(PARI) isok(n) = {if (((n % 2)==0) || isprime(n) || !issquarefree(n), return (0)); f = factor(n); for (i=1, #f~, d = f[i, 1]; if (type(2*(d-2+n/d)/(d-1)) != "t_INT", return(0)); ); return (1); } \\ Michel Marcus, Jul 12 2013

Crossrefs

Sequence in context: A020242 A020255 A367243 * A329759 A041428 A052226

Adjacent sequences: A180245 A180246 A180247 * A180249 A180250 A180251

Keyword

nonn

Author

William F. Sindelar (w_sindelar(AT)juno.com), Aug 19 2010

Extensions

Edited by the Associate Editors of the OEIS, Sep 04 2010

Status

approved