A292079 Composite numbers m such that 2^m - 1 has a single prime factor of the form k*m + 1.

4, 6, 8, 9, 12, 20, 24, 27, 33, 49, 69, 77, 145, 425, 447, 567

Offset: 1

Author

Michel Marcus, Sep 12 2017

Keywords

• nonn
• more

Comments

From Thomas Ordowski, Sep 12 2017: (Start)

Composite numbers m such that A182590(m) = 1.

Problem: are there infinitely many such numbers?

Note that this single prime factor p is the only primitive prime factor of 2^m - 1 for all such m except 6, i.e., the multiplicative order of 2 modulo p is m. (End)

After 567, the only numbers < 1200 that may possibly be terms are 961, 1037, 1111, and 1115. - Jon E. Schoenfield, Dec 03 2017

a(17) > 1206. - Amiram Eldar, Apr 01 2021

Crossrefs

Cf. A001265, A002326, A182590.

Programs

Mathematica

Select[Range@ 150, And[CompositeQ@ #, Function[{m, p}, Total@ Boole@ Map[Divisible[# - 1, m] &, p] == 1] @@ {#, FactorInteger[2^# - 1][[All, 1]]}] &] (* Michael De Vlieger, Dec 06 2017 *)

PARI

lista(nn) = forcomposite(n=1, nn, my(f = factor(2^n-1)); if (sum(k=1, #f~, ((f[k, 1]-1) % n)==0) == 1, print1(n, ", ")));

Extensions

Erroneous terms 841 and 1127 and possible (but unconfirmed, and not necessarily next) term 1037 deleted by Jon E. Schoenfield, Dec 03 2017