A265285 - OEIS

%I #53 May 02 2024 04:35:45

%S 46657,2433601,67371265,351596817937,422240040001,18677955240001,

%T 458631349862401,286245437364810001,20717489165917230086401

%N Carmichael numbers (A002997) k such that k-1 is a square.

%C This sequence contains all Carmichael numbers n such that for all primes p dividing n, p-1 divides n-1 and furthermore, n-1 is a square.

%C Numbers sqrt(a(n)-1) form a subsequence of A135590. - _Max Alekseyev_, Apr 25 2024

%H G. Tarry, I. Franel, A. Korselt, and G. Vacca, <a href="https://oeis.org/wiki/File:Probl%C3%A8me_chinois.pdf">Problème chinois</a>, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.

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

%H <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.

%e 46657 is a term because 46657 - 1 = 46656 = 216^2.

%e 2433601 is a term because 2433601 - 1 = 2433600 = 1560^2.

%p isA002997:= proc(n) local F,p;

%p          if n::even or isprime(n)  then return false fi;

%p          F:= ifactors(n)[2];

%p          if max(seq(f[2],f=F)) > 1 then return false fi;

%p          andmap(f -> (n-1) mod (f[1]-1) = 0,  F)

%p end proc:

%p select(isA002997, [seq(4*i^2+1,i=1..10^6)]); # _Robert Israel_, Dec 08 2015

%o (PARI) is_c(n) = { my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1 }

%o for(n=1, 1e10, if(is_c(n) && issquare(n-1), print1(n, ", ")))

%o (PARI) lista(kmax) = {my(m); for(k = 2, kmax, m = k^2 + 1; if(!isprime(m), f = factor(k); for(i = 1, #f~, f[i, 2] *= 2); fordiv(f, d, if(!(m % (d+1)) && isprime(d+1), m /= (d+1))); if(m == 1, print1(k^2 + 1, ", ")))); } \\ _Amiram Eldar_, May 02 2024

%Y Subsequence of A265237 and of A265328.

%Y Cf. A002997, A135590, A265237, A303791.

%K nonn,hard,more

%O 1,1

%A _Altug Alkan_, Dec 06 2015

%E a(4)-a(5), using A002997 b-file, from _Michel Marcus_, Dec 07 2015

%E a(6) and a(7) from _Robert Israel_, Dec 08 2015

%E a(8) from _Max Alekseyev_, Apr 30 2018

%E a(9) from _Daniel Suteu_ confirmed by _Max Alekseyev_, Apr 25 2024