%I A127093
%S A127093 1,1,2,1,0,3,1,2,0,4,1,0,0,0,5,1,2,3,0,0,6,1,0,0,0,0,0,7,1,2,0,4,0,0,0,
%T A127093 8,1,0,3,0,0,0,0,0,9,1,2,0,0,5,0,0,0,0,10,1,0,0,0,0,0,0,0,0,0,11,1,2,3,
%U A127093 4,0,6,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,0,0,13,1,2,0,0,0,0,7,0,0,0,0,0
%N A127093 Triangle read by rows: T(n,k)=k if k is a divisor of n; otherwise, T(n,
               k)=0 (1<=k<=n).
%C A127093 Sum of terms in row n = sigma(n) (sum of divisors of n, A000203(n)).
%C A127093 Euler's derivation of A127093 in polynomial form is in his proof of the 
               formula for Sigma(n): (let S=Sigma, then Euler proved that S(n) = 
               S(n-1) + S(n-2) - S(n-5) - S(n-7) + S(n-12) + S(n-15) - S(n-22) - 
               S(n-26),...).
%C A127093 [Young, p. 365-366], Euler begins, s = (1-x)*(1-x^2)*(1-x^3)...= 1 - 
               x - x^2 + x^5 + x^7 - x^12...; log s = log(1-x) + log(1-x^2) + log(1-x^3)...; 
               differentiating and then changing signs, Euler has t = x/(1-x) + 
               2x^2/(1-x^2) + 3x^3/(1-x^3) + 4x^4/(1-x^4) + 5x^5/(1-x^5)...
%C A127093 Finally, Euler expands each term of t into a geometric series, getting 
               A127093 in polynomial form: t =
%C A127093 x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + ...
%C A127093 ..+2x^2..........+ 2x^4..........+ 2x^6.........+ 2x^8 + ...
%C A127093 .........+ 3x^3..................+ 3x^6.................+ ...
%C A127093 .................+ 4x^4.........................+ 4x^8..+ ...
%C A127093 .........................+ 5x^5.........................+ ...
%C A127093 .................................+ 6x^6.................+ ...
%C A127093 .........................................+ 7x^7.........+ ...
%C A127093 ................................................+ 8x^8..+ ...
%D A127093 David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John 
               Wiley & Sons, 2005, appendix.
%D A127093 L. Euler, "Discovery of a Most Extraordinary Law of the Numbers Concerning 
               the Sum of Their Divisors"; pp. 358-367 of Robert M. Young, "Excursions 
               in Calculus, An Interplay of the Continuous and the Discrete", MAA, 
               1992. See p. 366.
%F A127093 k-th column is composed of "k" interspersed with (k-1) zeros.
%F A127093 Let M = A127093 as an infinite lower triangular matrix and V = the harmonic 
               series as a vector: [1/1, 1/2, 1/3,...]. then M*V = d(n), A000005: 
               [1, 2, 2, 3, 2, 4, 2, 4, 3, 4,...]. M^2 * V = A060640: [1, 5, 7, 
               17, 11, 35, 15, 49, 34, 55,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 10 2007
%F A127093 Also mod(n-1;k) - mod(n;k) + 1 (1<=k<=n) - Mats Granvik (mgranvik(AT)abo.fi), 
               Aug 31 2007
%e A127093 T(8,4) = 4 since 4 divides 8.
%e A127093 T(9,3) = 3 since 3 divides 9.
%e A127093 First few rows of the triangle are:
%e A127093 1;
%e A127093 1, 2;
%e A127093 1, 0, 3;
%e A127093 1, 2, 0, 4;
%e A127093 1, 0, 0, 0, 5;
%e A127093 1, 2, 3, 0, 0, 6;
%e A127093 1, 0, 0, 0, 0, 0, 7;
%e A127093 1, 2, 0, 4, 0, 0, 0, 8;
%e A127093 1, 0, 3, 0, 0, 0, 0, 0, 9;
%e A127093 ...
%p A127093 T:=proc(n,k) if type(n/k, integer)=true then k else 0 fi end: for n from 
               1 to 16 do seq(T(n,k),k=1..n) od; # yields sequence in triangular 
               form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 20 2007
%o A127093 (Excel cell formula) mod(row()-1;column()) - mod(row();column()) + 1 
               - Mats Granvik (mgranvik(AT)abo.fi), Aug 31 2007
%Y A127093 Reversal = A127094
%Y A127093 Cf. A127094, A127095, A127096, A127097, A127098, A127099, A000203, A126988, 
               A127013, A127057, A038040, A024916, A060640, A001001.
%Y A127093 Cf. A000005, A060640.
%Y A127093 Sequence in context: A143256 A143151 A130106 this_sequence A141543 A146540 
               A162922
%Y A127093 Adjacent sequences: A127090 A127091 A127092 this_sequence A127094 A127095 
               A127096
%K A127093 nonn,tabl
%O A127093 1,3
%A A127093 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 05 2007, Apr 04 2007
%E A127093 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 20 2007