A265285 Carmichael numbers (A002997) k such that k-1 is a square.

46657, 2433601, 67371265, 351596817937, 422240040001, 18677955240001, 458631349862401, 286245437364810001, 20717489165917230086401

Offset: 1

Author

Altug Alkan, Dec 06 2015

Keywords

• nonn
• hard
• more

Comments

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.

Numbers sqrt(a(n)-1) form a subsequence of A135590. - Max Alekseyev, Apr 25 2024

Examples

			46657 is a term because 46657 - 1 = 46656 = 216^2.
2433601 is a term because 2433601 - 1 = 2433600 = 1560^2.
		

Links

• G. Tarry, I. Franel, A. Korselt, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
• Eric Weisstein's World of Mathematics, Carmichael Number.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Subsequence of A265237 and of A265328.

Cf. A002997, A135590, A265237, A303791.

Programs

Maple

isA002997:= proc(n) local F,p;
         if n::even or isprime(n)  then return false fi;
         F:= ifactors(n)[2];
         if max(seq(f[2],f=F)) > 1 then return false fi;
         andmap(f -> (n-1) mod (f[1]-1) = 0,  F)
end proc:
select(isA002997, [seq(4*i^2+1,i=1..10^6)]); # Robert Israel, Dec 08 2015

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

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

Extensions

a(4)-a(5), using A002997 b-file, from Michel Marcus, Dec 07 2015

a(6) and a(7) from Robert Israel, Dec 08 2015

a(8) from Max Alekseyev, Apr 30 2018

a(9) from Daniel Suteu confirmed by Max Alekseyev, Apr 25 2024