

A286942


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


2



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, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173
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OFFSET

1,2


COMMENTS

From Michael De Vlieger, May 18 2017: (Start)
Row n of a(n) is the list of numbers 1 <= k <= A002110(n) that are coprime to A002110(n1).
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).
Row lengths = A005867(n) + A005867(n1): {2, 3, 10, 56, 528, 6240, 97920, ...}.
1 is coprime to all n thus delimits the rows of a(n).
The smallest prime q in row n of a(n) is gpf(primorial(n)) = A006530(A002110(n)) = prime(n) by definition of primorial.
The smallest composite x in row n of a(n) is q^2 = A001248(n).
The Kummer number A057588(n) = A002110(n)  1 is the largest term in row n of a(n). (End)


LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..6839 (rows 1 <= n <= 6).
Eric Weisstein's World of Mathematics, Relatively Prime  Michael De Vlieger, May 18 2017
Seqfan, Formula for the sequence.


FORMULA

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


EXAMPLE

The triangle starts:
1, 2;
1, 3, 5;
1, 5, 7, 11, 13, 17, 19, 23, 25, 29
Example1:
To find row n of the irregular triangle A286942, take a running sum for each value in the irregular triangle row n1 of A286941 with A002110(n1) b1 times, where b is the largest prime factor in A002110(n).
For example to find row 3 of A286942: Take a running sum for both 1 and 5 in row n1 of A286941 with A002110(31)=6, 51=4 times, where b is the largest prime factor 5 in A002110(3).
Result:
1 5
7 11
13 17
19 23
25 29
Equal to row 3 of A286942: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29.
Example2:
To find row n of the irregular triangle A279864, multiply each value in row n1 of A286941 with the largest prime factor b in A002110(n).
Example for n=3: b=5.
1*5=5
5*5=25
Example3:
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.
For n=3.
A286942 row n = 1, 5, 7, 11, 13, 17, 19, 23, 25, 29.
A279864 row n = 5, 25.
Removing values 5, 25 from the values in A286942 row n gives row n of A286941: 1, 7, 11, 13, 17, 19, 23, 29.


MATHEMATICA

Table[Select[Range@ #2, Function[k, CoprimeQ[k, #1]]] & @@ Map[Times @@ # &, {Most@ #, #}] &@ Prime@ Range@ n, {n, 4}] // Flatten (* Michael De Vlieger, May 18 2017 *)


CROSSREFS

Cf. A002110, A005867, A279864, A286941, A286942.
Sequence in context: A297519 A297749 A173588 * A125076 A220562 A215564
Adjacent sequences: A286939 A286940 A286941 * A286943 A286944 A286945


KEYWORD

nonn,tabf


AUTHOR

Jamie Morken and Michael De Vlieger, May 16 2017


EXTENSIONS

More terms from Michael De Vlieger, May 18 2017


STATUS

approved



