This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A127093 #117 Apr 05 2025 10:24:24
%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,
%T A127093 0,8,1,0,3,0,0,0,0,0,9,1,2,0,0,5,0,0,0,0,10,1,0,0,0,0,0,0,0,0,0,11,1,
%U A127093 2,3,4,0,6,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,0,0,13,1,2,0,0,0,0,7,0,0,0,0,0,0,14
%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).
%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, pp. 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 + ...
%C A127093 T(n,k) is the sum of all the k-th roots of unity each raised to the n-th power. - _Geoffrey Critzer_, Jan 02 2016
%C A127093 From _Davis Smith_, Mar 11 2019: (Start)
%C A127093 For n > 1, A020639(n) is the leftmost term, other than 0 or 1, in the n-th row of this array. As mentioned in the Formula section, the k-th column is period k: repeat [k, 0, 0, ..., 0], but this also means that it's the characteristic function of the multiples of k multiplied by k. T(n,1) = A000012(n), T(n,2) = 2*A059841(n), T(n,3) = 3*A079978(n), T(n,4) = 4*A121262(n), T(n,5) = 5*A079998(n), and so on.
%C A127093 The terms in the n-th row, other than 0, are the factors of n. If n > 1 and for every k, 1 <= k < n, T(n,k) = 0 or 1, then n is prime. (End)
%C A127093 From _Gary W. Adamson_, Aug 07 2019: (Start)
%C A127093 Row terms of the triangle can be used to calculate E(n) in A002654): (1, 1, 0, 1, 2, 0, 0, 1, 1, 2, ...), and A004018, the number of points in a square lattice on the circle of radius sqrt(n), A004018: (1, 4, 4, 0, 4, 8, 0, 0, 4, ...).
%C A127093 As to row terms in the triangle, let E(n) of even terms = 0,
%C A127093 E(integers of the form 4*k - 1 = (-1), and E(integers of the form 4*k + 1 = 1.
%C A127093 Then E(n) is the sum of the E(n)'s of the factors of n in the triangle rows. Example: E(10) = Sum: ((E(1) + E(2) + E(5) + E(10)) = ((1 + 0 + 1 + 0) = 2, matching A002654(10).
%C A127093 To get A004018, multiply the result by 4, getting A004018(10) = 8.
%C A127093 The total numbers of lattice points = 4r^2 = E(1) + ((E(2))/2 + ((E(3))/3 + ((E(4))/4 + ((E(5))/5 + .... Since E(even integers) are zero, E(integers of the form (4*k - 1)) = (-1), and E(integers of the form (4*k + 1)) = (+1); we are left with 4r^2 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ..., which is approximately equal to Pi(r^2). (End)
%C A127093 T(n,k) is also the number of parts in the partition of n into k equal parts. - _Omar E. Pol_, May 05 2020
%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.
%H A127093 Reinhard Zumkeller, <a href="/A127093/b127093.txt">Rows n = 1..100 of triangle, flattened</a>
%H A127093 Grant Sanderson, <a href="https://www.3blue1brown.com/lessons/leibniz-formula">Pi hiding in prime regularities</a>
%H A127093 Leonhard Euler, <a href="http://eulerarchive.maa.org/pages/E175.html">Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs</a>, 1747, The Euler Archive, (Eneström Index) E175.
%H A127093 Leonhard Euler, <a href="http://eulerarchive.maa.org//pages/E243.html">Observatio de summis divisorum</a>
%H A127093 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Divisor.html">Divisor</a>
%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_, May 10 2007
%F A127093 T(n,k) = ((n-1) mod k) - (n mod k) + 1 (1 <= k <= n). - _Mats Granvik_, Aug 31 2007
%F A127093 T(n,k) = k * 0^(n mod k). - _Reinhard Zumkeller_, Jan 15 2011
%F A127093 G.f.: Sum_{k>=1} k * x^k * y^k/(1-x^k) = Sum_{m>=1} x^m * y/(1 - x^m*y)^2. - _Robert Israel_, Aug 08 2016
%F A127093 T(n,k) = Sum_{d|k} mu(k/d)*sigma(gcd(n,d)). - _Ridouane Oudra_, Apr 05 2025
%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:
%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 A127093:=proc(n,k) if type(n/k, integer)=true then k else 0 fi end:
%p A127093 for n from 1 to 16 do seq(A127093(n,k),k=1..n) od; # yields sequence in triangular form - _Emeric Deutsch_, Jan 20 2007
%t A127093 t[n_, k_] := k*Boole[Divisible[n, k]]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 17 2014 *)
%t A127093 Table[ SeriesCoefficient[k*x^k/(1 - x^k), {x, 0, n}], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 14 2015 *)
%o A127093 (Excel) mod(row()-1;column()) - mod(row();column()) + 1 - _Mats Granvik_, Aug 31 2007
%o A127093 (Haskell)
%o A127093 a127093 n k = a127093_row n !! (k-1)
%o A127093 a127093_row n = zipWith (*) [1..n] $ map ((0 ^) . (mod n)) [1..n]
%o A127093 a127093_tabl = map a127093_row [1..]
%o A127093 -- _Reinhard Zumkeller_, Jan 15 2011
%o A127093 (PARI) trianglerows(n) = for(x=1, n, for(k=1, x, if(x%k==0, print1(k, ", "), print1("0, "))); print(""))
%o A127093 /* Print initial 9 rows of triangle as follows: */
%o A127093 trianglerows(9) \\ _Felix Fröhlich_, Mar 26 2019
%Y A127093 Reversal = A127094
%Y A127093 Cf. A127094, A123229, A127096, A127097, A127098, A127099, A000203, A126988, A127013, A127057, A038040, A024916, A060640, A001001.
%Y A127093 Cf. A000005, A060640.
%Y A127093 Cf. A027750.
%Y A127093 Cf. A000012 (the first column), A020639, A059841 (the second column when multiplied by 2), A079978 (the third column when multiplied by 2), A079998 (the fifth column when multiplied by 5), A121262 (the fourth column when multiplied by 4).
%Y A127093 Cf. A002654, A004018, A008683, A050873.
%K A127093 nonn,easy,tabl
%O A127093 1,3
%A A127093 _Gary W. Adamson_, Jan 05 2007, Apr 04 2007