A331118 Irregular triangle read by rows where row n lists primitive first differences in the reduced residue system of A002110(n).

2, 2, 4, 2, 4, 6, 2, 4, 6, 8, 10, 2, 4, 6, 8, 10, 12, 14, 2, 4, 6, 8, 10, 12, 14, 16, 18, 22, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 34, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36

Offset: 1

Author

Michael De Vlieger, Jan 10 2020

Keywords

• nonn
• tabf

Comments

Let primorial P(n) = A002110(n) and let r < P(n) be a number such that gcd(r, P(n)) = 1. Thus r is a residue in the reduced residue system (RRS) of P(n), and the number of r pertaining to P(n) is given by phi(P(n)) = A005867(n). We take the union of the first differences of the r in the RRS of P(n) to arrive at row n of this sequence.

Let L be the run length of numbers m in the cototient of a number k and let the first differences D in the RRS of k. The cototient includes any m such that at least 1 prime p | m also divides k, in other words, any m such that gcd(m, k) > 1. We note L = D - 1.

Row n of this sequence is the union of first differences of row n of A286941.

Let D be a primitive first difference as defined above. D is necessarily even since P(n) (for n > 0) is even and all r are odd.

Length of row n = A329815(n).

Examples

			Triangle begins:
n    Row
1    2;
2    2, 4;
3    2, 4, 6;
4    2, 4, 6, 8, 10;
5    2, 4, 6, 8, 10, 12, 14;
6    2, 4, 6, 8, 10, 12, 14, 16, 18,     22;
7    2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26;
8    2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30,     34;
...
(Triangle is organized so that the D appear in columns.)
Row 1 = {2} because P(1) = 2 is prime and has only 2 itself in the cototient.
Row 2 = {2, 4} since the numbers {1, 5} are coprime to P(2) = 6, and their difference is 4.
Row 3 contains {2, 4, 6} since we encounter the run lengths 6 between 1 and 7, 4 between 7 and 11, and 2 between 11 and 13. The run lengths are repeated but no new lengths appear for P(3) = 30.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..5275 (rows 1 <= n <= 44, flattened)
• Mario Ziller, On differences between consecutive numbers coprime to primorials, arXiv:2007.01808 [math.NT], 2020.

Crossrefs

Cf. A002110, A005867, A048670, A286941, A329815.

Programs

Mathematica

Table[Block[{r = 1, s = {}}, Do[If[GCD[i, P] == 1, If[FreeQ[s, #], AppendTo[s, #]] &[i - r]; r = i], {i, 3, P/If[P > 6, 2, 1/2], 2}]; Union@ s], {P, FoldList[Times, Prime@ Range@ 8]}] // Flatten

Formula

A048670(n) = largest term in row n.

A329815(n) = length of row n.