A324975 Rank of the n-th Carmichael number.

6, 10, 12, 8, 8, 10, 6, 6, 8, 18, 52, 12, 12, 18, 98, 164, 22, 6, 50, 8, 96, 34, 52, 46, 52, 6, 6, 156, 20, 46, 36, 32, 16, 8, 304, 36, 20, 36, 10, 316, 76, 468, 8, 30, 24, 1580, 84, 54, 8, 12, 250, 28, 92, 36, 20, 418, 456, 928, 188, 16, 8, 276, 284, 56, 144

Offset: 1

Author

Bernd C. Kellner and Jonathan Sondow, Mar 24 2019

Keywords

• nonn

Comments

See A324974 for definition and explanation of rank of a special polygonal number, hence of rank of a Carmichael number A002997 by Kellner and Sondow 2019.

The ranks of the primary Carmichael numbers A324316 form the subsequence A324976.

Examples

			If m = A002997(1) = 561 = 3*11*17, then p = 17, so a(1) = 2+2*((561/17)-1)/(17-1) = 6.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
• Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
• Wikipedia, Polygonal number

Crossrefs

Subsequence of A324974.

A324976 is a subsequence.

Cf. also A002997, A324316, A324972, A324973, A324977.

Programs

Mathematica

T = Cases[Range[1, 10000000, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]];
GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
Table[2 + 2*(T[[i]]/GPF[T[[i]]] - 1)/(GPF[T[[i]]] - 1), {i, Length[T]}]

Formula

a(n) = 2+2*((m/p)-1)/(p-1), where m = A002997(n) and p is its greatest prime factor. (See Formula in A324974.) Hence a(n) is even, by Carmichael's theorem that p-1 divides (m/p)-1, for any prime factor p of a Carmichael number m.