A216928 - OEIS

%I #31 Feb 24 2023 19:33:23

%S 399,8855,588455,139501439,3512071871,199195047359,14563696180319,

%T 989565001538399,20576473996736735,4049149795181043839,

%U 409810997884396741919,46852073639840281125599,6414735508880546179805759,466807799396932243821123839,41222773167337486494297521279

%N Least Lucas-Carmichael number with n prime factors.

%C Is this sequence infinite? - _Charles R Greathouse IV_, Sep 23 2012

%C a(15) <= 6414735508880546179805759. a(16) <= 466807799396932243821123839. - _Donovan Johnson_, Sep 26 2012

%H Daniel Suteu, <a href="/A216928/b216928.txt">Table of n, a(n) for n = 3..35</a>

%H Ed Copeland and Brady Haran, <a href="https://www.youtube.com/watch?v=yfr3BIk6KFc">Something special about 399</a>, Numberphile video (2015).

%o (PARI)

%o lucas_carmichael(A, B, k) = A=max(A, vecprod(primes(k+1))\2); (f(m, l, lo, k) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, lo=max(lo, ceil(A/m)); my(t=lift(-1/Mod(m,l))); while(t < lo, t += l); forstep(p=t, hi, l, if(isprime(p), my(n=m*p); if((n+1)%(p+1) == 0, listput(list, n)))), forprime(p=lo, hi, if(gcd(m, p+1) == 1, list=concat(list, f(m*p, lcm(l, p+1), p+1, k-1))))); list); vecsort(Vec(f(1, 1, 3, k)));

%o a(n) = if(n < 3, return()); my(x=vecprod(primes(n+1))\2,y=2*x); while(1, my(v=lucas_carmichael(x,y,n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ _Daniel Suteu_, Feb 24 2023

%Y Cf. A006972 (Lucas-Carmichael numbers).

%K nonn

%O 3,1

%A _Tim Johannes Ohrtmann_, Sep 20 2012

%E a(7)-a(12) from _Donovan Johnson_, Sep 22 2012

%E a(13)-a(14) from _Donovan Johnson_, Sep 26 2012

%E a(15)-a(16) confirmed and a(17) added by _Daniel Suteu_, Aug 29 2022