A265628 Carmichael numbers (A002997) of the form k^3 + 1.

1729, 46657, 2628073, 19683001, 110592000001, 432081216001, 2116874304001, 3176523000001, 312328165704192001, 12062716067698821000001, 211215936967181638848001, 411354705193473163968001, 14295706553536348081491001, 32490089562753934948660824001

Offset: 1

Author

Altug Alkan, Dec 10 2015

Keywords

• nonn
• hard
• more

Comments

For the first nine Carmichael numbers of the form k^3 + 1, the values of k + 1 are 13, 37, 139, 271, 4801, 7561, 12841, 14701, 678481 and only 14701 is not a prime number.

The sequence also includes: 32490089562753934948660824001, 782293837499544845175052968001, 611009032634107957276386802479001. - Daniel Suteu, Dec 25 2020

Examples

			2628073 is a term because it is a Carmichael number and 2628073 = 138^3 + 1.
		

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

Cf. A002997, A265285, A265328.

Programs

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(k=n^3+1), print1(k, ", ")))

PARI

lista(kmax) = {my(m); for(k = 2, kmax, f = factor(k); for(i = 1, #f~, f[i, 2] *= 3); m = k^3 + 1; fordiv(f, d, if(!(m % (d+1)) && isprime(d+1), m /= (d+1))); if(m == 1, print1(k^3 + 1, ", "))); } \\ Amiram Eldar, May 02 2024

Extensions

More terms from Alois P. Heinz, Dec 10 2015

a(10)-a(13) from Daniel Suteu, Dec 25 2020

a(14) from Daniel Suteu confirmed by Amiram Eldar, May 02 2024