login
Carmichael numbers divisible by 7 and 17.
1

%I #28 Jul 29 2017 12:07:09

%S 825265,6840001,16778881,47006785,413631505,490503601,547652161,

%T 1180398961,1529544961,1597009393,2265650401,2313774001,2523947041,

%U 2560104001,2586927553,3180632833,3754483201,4477793761,5106068065,5394826801,6218177329,6453043345,8053562881,10152380161

%N Carmichael numbers divisible by 7 and 17.

%C Conjecture (1): Any Carmichael number C divisible by 7 and 17 of the form 10k+1 can be written as C = 7*17*(120n+119). We got for the tested numbers the following values for n: 478, 1174, 34348, 38350, 82660, 107110, 158658, 162028, 176746, 179278, 262918, 313570, 377788, 563974, 710950.

%C Conjecture (2): Any Carmichael number C divisible by 7 and 17 of the form 10k+5 can be written as C = 7*17*(120n+95). We got for the tested numbers the following values for n: 57, 3291, 28965, 357567, 451893.

%C Conjecture (3): Any Carmichael number C divisible by 7 and 17 of the form 10k+3 can be written as C = 7*17*(120n+47). We got for the tested numbers the following values for n: 111835, 181157, 222733.

%C Conjecture (4): Any Carmichael number C divisible by 7 and 17 of the form 10k+9 can be written as C = 7*17*(120n+71). We got for the tested number the following value for n: 435446.

%C Note: the property of being factorizable as C = p*q*(n*(p*q+1) + p*q), where p and q are prime, is not derived from Korselt's criterion, and is not shared by all Carmichael numbers; e.g., for the Carmichael number 340561 = 13*17*23*67, even though it is of the form 10k+1, we have (23*67) mod (13*17+1) = 209 != 221 = 13*17.

%H Charles R Greathouse IV, <a href="/A182532/b182532.txt">Table of n, a(n) for n = 1..10000</a>

%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>

%o (PARI) Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1

%o list(lim)=my(v=List()); forstep(n=825265, lim, 5712, if(Korselt(n), listput(v, n))); Vec(v)

%K nonn

%O 1,1

%A _Marius Coman_, May 04 2012

%E a(15) corrected by _Charles R Greathouse IV_, Oct 02 2012