OFFSET
1,1
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).
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.
EXAMPLE
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.
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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Michael De Vlieger, Jan 10 2020
STATUS
approved