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A309497 Irregular triangle read by rows: T(n,k) = A060753(n)*k-A038110(n)*A286941(n,k).

%I A309497 #34 Feb 16 2025 08:33:55
%S A309497 0,1,2,1,11,2,1,8,7,14,13,4,27,-18,1,4,23,26,13,32,19,22,41,44,31,18,
%T A309497 37,24,27,46,33,36,23,-6,-3,16,19,38,41,12,-1,2,-11,8,11,-2,17,4,-9,
%U A309497 -6,13,16,3,22,9,12,31,34,53,8
%N A309497 Irregular triangle read by rows: T(n,k) = A060753(n)*k-A038110(n)*A286941(n,k).
%C A309497 The sequence is Primorial rows of A308121.
%C A309497 Row n has length A005867(n).
%C A309497 Row n > 1 average value = A060753(n)/2.
%C A309497 Row n > 1 has sum = A002110(n-1)*A038110(n)/2.
%C A309497 First value on row(n) = A161527(n-1).
%C A309497 Last value on row(n) = A038110(n) for n > 2.
%C A309497 For n > 1, A060753(n) = Max(row) + Min(row).
%C A309497 For values x and y on row n > 1 at positions a and b on the row:
%C A309497 x + y = A060753(n), where a = A005867(n-1) - (b-1).
%C A309497 For n > 2 the penultimate value on row A002110(n) is given by
%C A309497   A038110(n)*A000040(n)-A060753(n).
%C A309497 Related identity:
%C A309497   A038110(n)/A038111(n)*(Prime(n)^2) - (A038110(n)/A038111(n)*((A038110(n)*Prime(n) - A060753(n))*Prime(n)/A038110(n))) = 1.
%H A309497 Michael De Vlieger, <a href="/A309497/b309497.txt">Table of n, a(n) for n = 0..6299</a> (rows n = 0..6, flattened)
%H A309497 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Totative.html">Totative</a>
%e A309497 The triangle starts:
%e A309497 row1: 0;
%e A309497 row2: 1;
%e A309497 row3: 2, 1;
%e A309497 row4: 11, 2, 1, 8, 7, 14, 13, 4;
%e A309497 row5: 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8;
%t A309497 row[0] = 0; row[n_] := -(v = Numerator[Product[1 - 1/Prime[i], {i, 1, n}] / Prime[n]] * Select[Range[(p = Product[Prime[i], {i, 1, n}])], CoprimeQ[p, #] &]) + Denominator[Product[((pr = Prime[i]) - 1)/pr, {i, 1, n}]] * Range[Length[v]]; Table[row[n], {n, 0, 4}] // Flatten (* _Amiram Eldar_, Aug 10 2019 *)
%Y A309497 Cf. A058250, A005867, A002110, A038110, A038111, A060753, A286941, A058262, A161527, A083140, A308121.
%K A309497 sign,look,tabf
%O A309497 0,3
%A A309497 _Jamie Morken_, Aug 05 2019