A331118 - OEIS

%I #18 Jul 13 2020 21:39:55

%S 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,

%T 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,

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

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

%C 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.

%C 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.

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

%C 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.

%C Length of row n = A329815(n).

%H Michael De Vlieger, <a href="/A331118/b331118.txt">Table of n, a(n) for n = 1..5275</a> (rows 1 <= n <= 44, flattened)

%H Mario Ziller, <a href="https://arxiv.org/abs/2007.01808">On differences between consecutive numbers coprime to primorials</a>, arXiv:2007.01808 [math.NT], 2020.

%F A048670(n) = largest term in row n.

%F A329815(n) = length of row n.

%e Triangle begins:

%e n    Row

%e 1    2;

%e 2    2, 4;

%e 3    2, 4, 6;

%e 4    2, 4, 6, 8, 10;

%e 5    2, 4, 6, 8, 10, 12, 14;

%e 6    2, 4, 6, 8, 10, 12, 14, 16, 18,     22;

%e 7    2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26;

%e 8    2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30,     34;

%e ...

%e (Triangle is organized so that the D appear in columns.)

%e Row 1 = {2} because P(1) = 2 is prime and has only 2 itself in the cototient.

%e Row 2 = {2, 4} since the numbers {1, 5} are coprime to P(2) = 6, and their difference is 4.

%e 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.

%t 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

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

%K nonn,tabf

%O 1,1

%A _Michael De Vlieger_, Jan 10 2020