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A133994 Irregular array read by rows: n-th row contains (in numerical order) both the positive integers <= n that are divisors of n and those that are coprime to n.

%I A133994 #19 Feb 09 2016 06:23:41
%S A133994 1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,2,3,5,6,1,2,3,4,5,6,7,1,2,3,4,5,7,8,
%T A133994 1,2,3,4,5,7,8,9,1,2,3,5,7,9,10,1,2,3,4,5,6,7,8,9,10,11,1,2,3,4,5,6,7,
%U A133994 11,12,1,2,3,4,5,6,7,8,9,10,11,12,13,1,2,3,5,7,9,11,13,14
%N A133994 Irregular array read by rows: n-th row contains (in numerical order) both the positive integers <= n that are divisors of n and those that are coprime to n.
%C A133994 Row n contains A073757(n) terms.
%C A133994 The number 1 would appear twice for each n >= 1 if one takes the union of the divisor list of n and the list of the smallest positive reduced residue system modulo n. - _Wolfdieter Lang_, Jan 16 2016
%H A133994 Robert Israel, <a href="/A133994/b133994.txt">Table of n, a(n) for n = 1..10003</a> (rows 1 to 174, flattened)
%e A133994 The divisors of 12 are: 1,2,3,4,6,12. The positive integers which are <= 12 and are coprime to 12 are: 1,5,7,11. So row 12 is the union of these two sets: 1,2,3,4,5,6,7,11,12.
%e A133994 The irregular triangle T(n, k) starts:
%e A133994 n\k 1 2 3 4 5 6  7  8  9 10 11 12 13 14 15 16 17
%e A133994 1:  1
%e A133994 2:  1 2
%e A133994 3:  1 2 3
%e A133994 4:  1 2 3 4
%e A133994 5:  1 2 3 4 5
%e A133994 6:  1 2 3 5 6
%e A133994 7:  1 2 3 4 5 6  7
%e A133994 8:  1 2 3 4 5 7  8
%e A133994 9:  1 2 3 4 5 7  8  9
%e A133994 10: 1 2 3 5 7 9 10
%e A133994 11: 1 2 3 4 5 6  7  8  9 10 11
%e A133994 12: 1 2 3 4 5 6  7 11 12
%e A133994 13: 1 2 3 4 5 6  7  8  9 10 11 12 13
%e A133994 14: 1 2 3 5 7 9 11 13 14
%e A133994 15: 1 2 3 4 5 7  8 11 13 14 15
%e A133994 16: 1 2 3 4 5 7  8  9 11 13 15 16
%e A133994 17: 1 2 3 4 5 6  7  8  9 10 11 12 13 14 15 16 17
%e A133994 18: 1 2 3 5 6 7  9 11 13 17 18 19
%e A133994 ... Formatted by _Wolfdieter Lang_, Jan 16 2016
%p A133994 row:= n -> op(select(t -> member(igcd(t,n), [1,t]), [$1..n])):
%p A133994 seq(row(n), n=1..30); # _Robert Israel_, Jan 18 2016
%t A133994 row[n_] := Divisors[n] ~Union~ Select[Range[n], CoprimeQ[n, #]&]; Array[ row, 15] // Flatten (* _Jean-François Alcover_, Jan 18 2016 *)
%Y A133994 Cf. A073757, A133995.
%K A133994 nonn,tabf
%O A133994 1,3
%A A133994 _Leroy Quet_, Oct 01 2007