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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). 45
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

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

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

Also mod(n-1;k) - mod(n;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 are:

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

CROSSREFS

Reversal = A127094

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

Cf. A000005, A060640.

Cf. A027750.

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 February 18 17:08 EST 2018. Contains 299325 sequences. (Running on oeis4.)