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A286942 Irregular triangle read by rows: numbers 1 <= k <= (A002110(n) - 1) where gcd(k, A002110(n - 1)) = 1.

%I A286942 #24 Feb 16 2025 08:33:45
%S A286942 1,2,1,3,5,1,5,7,11,13,17,19,23,25,29,1,7,11,13,17,19,23,29,31,37,41,
%T A286942 43,47,49,53,59,61,67,71,73,77,79,83,89,91,97,101,103,107,109,113,119,
%U A286942 121,127,131,133,137,139,143,149,151,157,161,163,167,169,173
%N A286942 Irregular triangle read by rows: numbers 1 <= k <= (A002110(n) - 1) where gcd(k, A002110(n - 1)) = 1.
%C A286942 From _Michael De Vlieger_, May 18 2017: (Start)
%C A286942 Row n of a(n) is the list of numbers 1 <= k <= A002110(n) that are coprime to A002110(n-1).
%C A286942 A286941(n) and A279864(n) are subsets of a(n) such that the terms of the rows of each sequence combined and sorted comprise all the terms of a(n).
%C A286942 Row lengths = A005867(n) + A005867(n-1): {2, 3, 10, 56, 528, 6240, 97920, ...}.
%C A286942 1 is coprime to all n thus delimits the rows of a(n).
%C A286942 The smallest prime q in row n of a(n) is gpf(primorial(n)) = A006530(A002110(n)) = prime(n) by definition of primorial.
%C A286942 The smallest composite x in row n of a(n) is q^2 = A001248(n).
%C A286942 The Kummer number A057588(n) = A002110(n) - 1 is the largest term in row n of a(n). (End)
%H A286942 Michael De Vlieger, <a href="/A286942/b286942.txt">Table of n, a(n) for n = 1..6839</a> (rows 1 <= n <= 6).
%H A286942 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RelativelyPrime.html">Relatively Prime</a> - _Michael De Vlieger_, May 18 2017
%H A286942 Seqfan, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2017-May/017552.htm">Formula for the sequence</a>.
%F A286942 a(n) = union(A286941(n), A279864(n)) where n consists of all terms in row n of each sequence. - _Michael De Vlieger_, May 18 2017
%e A286942 The triangle starts:
%e A286942 1, 2;
%e A286942 1, 3, 5;
%e A286942 1, 5, 7, 11, 13, 17, 19, 23, 25, 29
%e A286942 Example1:
%e A286942 To find row n of the irregular triangle A286942, take a running sum for each value in the irregular triangle row n-1 of A286941 with A002110(n-1) b-1 times, where b is the largest prime factor in A002110(n).
%e A286942 For example to find row 3 of A286942: Take a running sum for both 1 and 5 in row n-1 of A286941 with A002110(3-1)=6, 5-1=4 times, where b is the largest prime factor 5 in A002110(3).
%e A286942 Result:
%e A286942 1 5
%e A286942 7 11
%e A286942 13 17
%e A286942 19 23
%e A286942 25 29
%e A286942 Equal to row 3 of A286942: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29.
%e A286942 Example2:
%e A286942 To find row n of the irregular triangle A279864, multiply each value in row n-1 of A286941 with the largest prime factor b in A002110(n).
%e A286942 Example for n=3: b=5.
%e A286942 1*5=5
%e A286942 5*5=25
%e A286942 Example3:
%e A286942 To find row n of the irregular triangle A286941, remove the values that are in row n of the irregular triangle A279864 from the values that are in row n of the irregular triangle A286942.
%e A286942 For n=3.
%e A286942 A286942 row n = 1, 5, 7, 11, 13, 17, 19, 23, 25, 29.
%e A286942 A279864 row n = 5, 25.
%e A286942 Removing values 5, 25 from the values in A286942 row n gives row n of A286941: 1, 7, 11, 13, 17, 19, 23, 29.
%t A286942 Table[Select[Range@ #2, Function[k, CoprimeQ[k, #1]]] & @@ Map[Times @@ # &, {Most@ #, #}] &@ Prime@ Range@ n, {n, 4}] // Flatten (* _Michael De Vlieger_, May 18 2017 *)
%Y A286942 Cf. A002110, A005867, A279864, A286941, A286942.
%K A286942 nonn,tabf
%O A286942 1,2
%A A286942 _Jamie Morken_ and _Michael De Vlieger_, May 16 2017
%E A286942 More terms from _Michael De Vlieger_, May 18 2017