A265285 Carmichael numbers (A002997) k such that k-1 is a square.
46657, 2433601, 67371265, 351596817937, 422240040001, 18677955240001, 458631349862401, 286245437364810001, 20717489165917230086401
Offset: 1
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.
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
Comments