A324976 Rank of the n-th primary Carmichael number.

12, 8, 18, 12, 52, 52, 20, 32, 16, 54, 8, 36, 124, 34, 12, 72, 96, 26, 28, 76, 98, 1804, 108, 124, 18, 72, 172, 120, 10, 104, 32, 244, 130, 376, 18, 92, 780, 36, 172, 92, 284, 24, 198, 12, 244, 64, 234, 340, 100, 284, 24, 124, 44, 518, 364, 16, 82, 148, 8, 206

Offset: 1

Author

Bernd C. Kellner and Jonathan Sondow, Mar 24 2019

Keywords

• nonn
• base

Comments

See A324974 for definition and explanation of rank of a special polygonal number, hence of rank of a primary Carmichael number A324316 by Kellner and Sondow 2019.

Examples

			If m = A324316(1) = 1729 = 7*13*19, then p = 19, so a(1) = 2+2*((1729/19)-1)/(19-1) = 12.
		

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 A324975 (rank of the n-th Carmichael number A002997) and of A324974 (rank of the n-th special polygonal number A324973).

Cf. also A324316, A324972.

Programs

Mathematica

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestCP[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] == # &];
T = Select[Range[1, 10^7, 2], TestCP[#] &];
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 = A324316(n) and p is its greatest prime factor. Hence a(n) is even; see Formula in A324975.

Extensions

More terms from Amiram Eldar, Mar 27 2019