A265628 - OEIS

%I #37 May 02 2024 04:35:26

%S 1729,46657,2628073,19683001,110592000001,432081216001,2116874304001,

%T 3176523000001,312328165704192001,12062716067698821000001,

%U 211215936967181638848001,411354705193473163968001,14295706553536348081491001,32490089562753934948660824001

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

%C 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.

%C The sequence also includes: 32490089562753934948660824001, 782293837499544845175052968001, 611009032634107957276386802479001. - _Daniel Suteu_, Dec 25 2020

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

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

%o (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

%Y Cf. A002997, A265285, A265328.

%K nonn,hard,more

%O 1,1

%A _Altug Alkan_, Dec 10 2015

%E More terms from _Alois P. Heinz_, Dec 10 2015

%E a(10)-a(13) from _Daniel Suteu_, Dec 25 2020

%E a(14) from _Daniel Suteu_ confirmed by _Amiram Eldar_, May 02 2024