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A127093 Triangle read by rows: T(n,k)=k if k is a divisor of n; otherwise, T(n,k)=0 (1 <= k <= n). 47
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Sum of terms in row n = sigma(n) (sum of divisors of n).

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), ...).

[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) + ...

Finally, Euler expands each term of t into a geometric series, getting A127093 in polynomial form: t =

  x +  x^2 +  x^3 +  x^4 +  x^5 +  x^6 +  x^7 +  x^8 + ...

    + 2x^2        + 2x^4        + 2x^6        + 2x^8 + ...

           + 3x^3               + 3x^6               + ...

                  + 4x^4                      + 4x^8 + ...

                         + 5x^5                      + ...

                                + 6x^6               + ...

                                       + 7x^7        + ...

                                              + 8x^8 + ...

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

From Davis Smith, Mar 11 2019: (Start)

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.

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)

From Gary W. Adamson, Aug 07 2019: (Start)

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, ...).

As to row terms in the triangle, let E(n) of even terms = 0,

  E(integers of the form 4*k - 1 = (-1), and E(integers of the form 4*k + 1 = 1.

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).

To get A004018, multiply the result by 4, getting A004018(10) = 8.

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)

REFERENCES

David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John Wiley & Sons, 2005, appendix.

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.

LINKS

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Grant Sanderson, Pi hiding in prime regularities

L. Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, 1747, The Euler Archive, (Eneström Index) E175.

L. Euler, Observatio de summis divisorum

Eric Weisstein's World of Mathematics, Divisor

FORMULA

k-th column is composed of "k" interspersed with (k-1) zeros.

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

T(n,k) = ((n-1) mod k) - (n mod k) + 1 (1 <= k <= n). - Mats Granvik, Aug 31 2007

T(n,k) = k * 0^(n mod k). - Reinhard Zumkeller, Jan 15 2011

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

EXAMPLE

T(8,4) = 4 since 4 divides 8.

T(9,3) = 3 since 3 divides 9.

First few rows of the triangle:

  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, 8;

  1, 0, 3, 0, 0, 0, 0, 0, 9;

  ...

MAPLE

A127093:=proc(n, k) if type(n/k, integer)=true then k else 0 fi end:

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

MATHEMATICA

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 *)

Table[ SeriesCoefficient[k*x^k/(1 - x^k), {x, 0, n}], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 14 2015 *)

PROG

(Excel) mod(row()-1; column()) - mod(row(); column()) + 1 - Mats Granvik, Aug 31 2007

(Haskell)

a127093 n k = a127093_row n !! (k-1)

a127093_row n = zipWith (*) [1..n] $ map ((0 ^) . (mod n)) [1..n]

a127093_tabl = map a127093_row [1..]

-- Reinhard Zumkeller, Jan 15 2011

(PARI) trianglerows(n) = for(x=1, n, for(k=1, x, if(x%k==0, print1(k, ", "), print1("0, "))); print(""))

/* Print initial 9 rows of triangle as follows: */

trianglerows(9) \\ Felix Fröhlich, Mar 26 2019

CROSSREFS

Reversal = A127094

Cf. A127094, A123229, A127096, A127097, A127098, A127099, A000203, A126988, A127013, A127057, A038040, A024916, A060640, A001001.

Cf. A000005, A060640.

Cf. A027750.

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).

Cf. A002654, A004018.

Sequence in context: A143256 A143151 A130106 * A141543 A182720 A146540

Adjacent sequences:  A127090 A127091 A127092 * A127094 A127095 A127096

KEYWORD

nonn,easy,tabl

AUTHOR

Gary W. Adamson, Jan 05 2007, Apr 04 2007

STATUS

approved

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Last modified October 15 09:22 EDT 2019. Contains 328026 sequences. (Running on oeis4.)