A127073 - OEIS

%I #12 Feb 16 2025 08:33:04

%S 45,245,405,561,637,639,833,891,1105,1377,1576,1729,2465,2701,2821,

%T 3321,3645,4753,5589,6345,6517,6601,7885,8911,10365,10585,12005,13833,

%U 15841,17152,17265,18179,18721,21141,23552,25681,26411,29341,31213,31621

%N Composite numbers k that divide 3^k - 2^k - 1, excluding powers of 2, 3 and 7.

%C This sequence includes all the Carmichael numbers (A002997).

%C Prime p divides 3^p - 2^p - 1. Quotients (3^p - 2^p - 1)/p, where p = prime(n), are listed in A127071.

%C Numbers k such that k divides 3^k - 2^k - 1 are listed in A127072.

%C Pseudoprimes in A127072 include all the powers of the primes {2, 3, 7}.

%C Numbers k such that k^2 divides 3^k - 2^k - 1 are listed in A127074.

%C Numbers k such that k^3 divides 3^k - 2^k - 1 are 1, 4, 7, ...

%H Amiram Eldar, <a href="/A127073/b127073.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael number</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pseudoprime.html">Pseudoprime</a>.

%t Select[Select[Range[2^15],!PrimeQ[ # ]&&IntegerQ[(3^#-2^#-1)/# ]&],!IntegerQ[Log[2,# ]]&&!IntegerQ[Log[3,# ]]&&!IntegerQ[Log[7,# ]]&]

%Y Cf. A002997, A127071, A127072, A127074.

%K nonn

%O 1,1

%A _Alexander Adamchuk_, Jan 04 2007