A392847 Number of twelve-prime Carmichael numbers less than 10^n.

2, 56, 983, 10018, 81434, 547590

Offset

19,1

Links

Table of n, a(n) for n=19..24.

Richard Pinch, Tables relating to Carmichael numbers.

Andrew Shallue and Jonathan Webster, Algorithms for Carmichael numbers, arXiv:2506.09903 [math.NT], 2025-2026.

Index entries for sequences related to Carmichael numbers.

Example

There are two Carmichael numbers less than 10^19 that have 12 prime factors: 7156857700403137441 = 11 * 13 * 17 * 19 * 29 * 37 * 41 * 43 * 61 * 97 * 109 * 127 and 9219669366496075201 = 11 * 13 * 17 * 19 * 29 * 37 * 41 * 61 * 71 * 73 * 113 * 127. Therefore, a(19) = 2.

Prog

(PARI)
carmichael_count(A, B, k) = A=max(A, vecprod(primes(k+1))\2); local(f); (f = (m, l, lo, k) -> my(count=0); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(count)); 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((m*p-1)%(p-1) == 0 && isprime(p), count++)), forprime(p=lo, hi, if(gcd(m, p-1) == 1, count += f(m*p, lcm(l, p-1), p+1, k-1)))); count); f(1, 1, 3, k);
a(n) = carmichael_count(1, 10^n, 12); \\ Daniel Suteu, Mar 03 2026

Crossrefs

For k-prime Carmichael numbers up to 10^n for k = 3,4,...,11, see A132195, A174612, A174613, A174614, A174615, A174616, A174617, A299710, A299711.

Cf. A002997, A006931, A055553.

Sequence in context: A056046 A080313 A080268 * A224297 A193476 A193475

Adjacent sequences: A392844 A392845 A392846 * A392848 A392849 A392850

Keyword

nonn,hard,more

Author

Amiram Eldar, Jan 24 2026

Status

approved