A329815 Number of distinct terms in the first difference sequence of the reduced residue system of the n-th primorial.

0, 1, 3, 5, 7, 10, 13, 16, 20, 23, 29, 33, 37, 43, 49, 53, 59, 66, 75, 84, 92, 99, 108, 116, 127, 132, 140, 148, 156, 164, 174, 185, 193, 206, 215, 224, 235, 245, 255, 267, 275, 286, 297, 308

Offset: 1

Author

Jamie Morken, Nov 21 2019

Keywords

• nonn
• hard

Comments

This sequence is the number of distinct terms in the first difference sequence for rows n in A286941 and A309497.

Number of distinct terms listed in row n of A331118. - Michael De Vlieger, Jul 11 2020

Examples

			For n = 3, A002110(3) = 30, RRS = {1, 7, 11, 13, 17, 19, 23, 29}, dRRS = {6, 4, 2, 4, 2, 4, 6}, so a(3) = 3.
		

Links

• Mario Ziller, On differences between consecutive numbers coprime to primorials, arXiv:2007.01808 [math.NT], 2020.

Crossrefs

Cf. A061498, A048670, A309497, A286941, A331118.

Programs

Mathematica

Primorial[n_] := Times @@ Prime[Range[n]]; Table[Length@ Union@ Differences@ Select[Range@ Primorial[n], CoprimeQ[#, Primorial[n]] &], {n, 7}] (* after Michael De Vlieger Jul 15 2017 from A061498 *)

PARI

f(n) = {my(va = select(x->(gcd(n, x)==1), [1..n])); vd = vector(#va-1, k, va[k+1] - va[k]); #Set(vd); } \\ A061498
a(n) = f(prod(i=1, n, prime(i))); \\ Michel Marcus, Dec 19 2019

Formula

a(n) = A061498(A002110(n)).

a(n) <= A048670(n)/2.

Extensions

a(12)-a(44) from Jamie Morken, Jul 11 2020 (after Mario Ziller)