cp's OEIS Frontend

A308121 Irregular triangle read by rows: T(n,k) = A109395(n)*k-A076512(n)*A038566(n,k).

%I A308121 #47 Aug 05 2019 10:47:21
%S A308121 0,1,1,2,1,1,1,2,3,4,2,1,1,2,3,4,5,6,1,1,1,1,1,2,1,2,1,2,3,4,1,2,1,2,
%T A308121 3,4,5,6,7,8,9,10,2,1,2,1,1,2,3,4,5,6,7,8,9,10,11,12,4,5,6,1,2,3,7,14,
%U A308121 13,4,11,2,1,8
%N A308121 Irregular triangle read by rows: T(n,k) = A109395(n)*k-A076512(n)*A038566(n,k).
%C A308121 Row n has length A000010(n).
%C A308121 Row n > 1 has sum = n*A076512(n)/2.
%C A308121 First value on row(n) = A076511(n).
%C A308121 Last value on row(n) = A076512(n) for n > 1.
%C A308121 For n > 1, A109395(n) = Max(row) + Min(row).
%C A308121 For values x and y on row n > 1 at positions a and b on the row:
%C A308121   x + y = A109395(n), where a = A000010(n) - (b-1).
%C A308121 For n > 2 the penultimate value on row A002110(n) is given by
%C A308121   A038110(n)*A000040(n)-A060753(n).
%C A308121 From _Charlie Neder_, Jun 05 2019: (Start)
%C A308121 If p is a prime dividing n, then row p*n consists of p copies of row n.
%C A308121 Conjecture: If n is odd, then row 2n can be obtained from row n by interchanging the first and second halves. (End)
%H A308121 Jamie Morken, <a href="/A308121/b308121.txt">Table of n, a(n) for n = 1..13413</a> (Rows n = 1..210 of triangle, flattened)
%e A308121 The sequence as an irregular triangle:
%e A308121   n/k 1, 2, 3, 4, ...
%e A308121    1: 0
%e A308121    2: 1
%e A308121    3: 1, 2
%e A308121    4: 1, 1
%e A308121    5: 1, 2, 3, 4
%e A308121    6: 2, 1
%e A308121    7: 1, 2, 3, 4, 5, 6
%e A308121    8: 1, 1, 1, 1
%e A308121    9: 1, 2, 1, 2, 1, 2
%e A308121   10: 3, 4, 1, 2
%e A308121   11: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
%e A308121   12: 2, 1, 2, 1
%e A308121   13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
%e A308121   14: 4, 5, 6, 1, 2, 3
%e A308121   15: 7, 14, 13, 4, 11, 2, 1, 8
%e A308121   ...
%e A308121   Row sums: 0, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 60.
%e A308121 T(14,5) = A109395(14)*5 - A076512(14)*A038566(14,5) = 7*5 - 3*11 = 2.
%e A308121 T(210,2) = A109395(210)*2 - A076512(210)*A038566(210,2) = 35*2 - 8*11 = -18.
%t A308121 Flatten@ Table[With[{a = n/GCD[n, #], b = Numerator[#/n]}, MapIndexed[a First@ #2 - b #1 &, Flatten@ Position[GCD[Table[Mod[k, n], {k, n - 1}], n], 1] /. {} -> {1}]] &@ EulerPhi@ n, {n, 15}] (* _Michael De Vlieger_, Jun 06 2019 *)
%o A308121 (PARI) vtot(n) = select(x->(gcd(n, x)==1), vector(n, k, k));
%o A308121 row(n) = my(q = eulerphi(n)/n, v = vtot(n)); vector(#v, k, denominator(q)*k - numerator(q)*v[k]); \\ _Michel Marcus_, May 14 2019
%Y A308121 Cf. A000010, A109395, A038566, A076511, A076512.
%K A308121 sign,tabf
%O A308121 1,4
%A A308121 _Jamie Morken_, May 13 2019