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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 (list; graph; refs; listen; history; text; internal format)
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(n-1).

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(n-1): {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 n-1 of A286941 with A002110(n-1) b-1 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 n-1 of A286941 with A002110(3-1)=6, 5-1=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 n-1 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

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Last modified May 15 07:17 EDT 2021. Contains 343909 sequences. (Running on oeis4.)