A127099 -id:A127099 - cp's OEIS frontend

A127108 Triangle read by rows, A127099 * A000012.

1, 5, 2, 7, 3, 3, 17, 10, 4, 4, 11, 5, 5, 5, 5, 35, 23, 15, 6, 6, 6, 15, 7, 7, 7, 7, 7, 7, 49, 34, 20, 20, 8, 8, 8, 8, 34, 21, 21, 9, 9, 9, 9, 9, 9, 55, 37, 25, 25, 25, 10, 10, 10, 10, 10, 23, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 119, 91, 67, 46, 30, 30, 12, 12, 12, 12, 12, 12

Offset: 0

Gary W. Adamson, Jan 05 2007, Jul 27 2008

The operation A000012 * A127099 generates n-th row of the triangle by taking partial sums of n-th row of triangle A127099. Row 4 of A127099 (7, 6, 0, 4) becomes row 4 of A127108: (17, 10, 4, 4).
Row sums = A001001: (1, 7, 13, 35, 31, 91, ...).
Left column of the triangle = A060640: (1, 5, 7, 17, 11, 35, ...).

			First few rows of the triangle:
   1;
   5,  2;
   7,  3,  3;
  17, 10,  4,  4;
  11,  5,  5,  5,  5;
  35, 23, 15,  6,  6,  6;
  15,  7,  7,  7,  7,  7,  7;
  49, 34, 20, 20,  8,  8,  8,  8;
  34, 21, 21,  9,  9,  9,  9,  9,  9;
  55, 37, 25, 25, 25, 10, 10, 10, 10, 10;
  ...

Cf. A060640, A001001, A126988, A127093, A000203, A127094, A123229, A127096, A127098, A127099.

Triangle read by rows, A127099 * A000012.

Edited by N. J. A. Sloane, Aug 13 2008 at the suggestion of R. J. Mathar

A127093 Triangle read by rows: T(n,k)=k if k is a divisor of n; otherwise, T(n,k)=0 (1 <= k <= n).

Original entry on oeis.org

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

Offset: 1

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

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)
T(n,k) is also the number of parts in the partition of n into k equal parts. - Omar E. Pol, May 05 2020

			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;
  ...

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.

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Grant Sanderson, Pi hiding in prime regularities
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, 1747, The Euler Archive, (Eneström Index) E175.
Leonhard Euler, Observatio de summis divisorum
Eric Weisstein's World of Mathematics, Divisor

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, A008683, A050873.

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

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

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

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 *)

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

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
T(n,k) = Sum_{d|k} mu(k/d)*sigma(gcd(n,d)). - Ridouane Oudra, Apr 05 2025

A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2p_i + 3p_i^2 + ... + (e_i+1)*p_i^e_i).

Original entry on oeis.org

1, 5, 7, 17, 11, 35, 15, 49, 34, 55, 23, 119, 27, 75, 77, 129, 35, 170, 39, 187, 105, 115, 47, 343, 86, 135, 142, 255, 59, 385, 63, 321, 161, 175, 165, 578, 75, 195, 189, 539, 83, 525, 87, 391, 374, 235, 95, 903, 162, 430, 245, 459, 107, 710, 253, 735, 273, 295, 119

Offset: 1

N. J. A. Sloane, Apr 17 2001

Equals row sums of triangle A143313. - Gary W. Adamson, Aug 06 2008
Equals row sums of triangle A127099. - Gary W. Adamson, Jul 27 2008
Sum of the divisors d2 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

			a(4) = a(2^2) = 1 + (2)*(2) + (3)*(2^2) = 17;
a(6) = a(2)*a(3) = (1 + (2)*(2))*(1+(2)*(3)) = (5)*(7) = 35.
a(6) = tau(1) + 2*tau(2) + 3*tau(3) + 6*tau(6) = 1 + 2*2 + 3*2 + 6*4 = 35.

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A000203, A001001, A006171, A038040 (Mobius transform), A049060, A057660, A057723, A327960 (Dirichlet inverse).

Cf. also triangles A027750, A127099, A143313.

a060640 n = sum [d * a000005 d | d <- a027750_row n]
-- Reinhard Zumkeller, Feb 29 2012

A060640 := proc(n) local ans, i, j; ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(1+sum((j+1)*ifactors(n)[2][i][1]^j,j=1..ifactors(n)[2][i][2])): od: RETURN(ans) end:

a[n_] := Total[#*DivisorSigma[1, n/#] & /@ Divisors[n]];
a /@ Range[59] (* Jean-François Alcover, May 19 2011, after Vladeta Jovovic *)
f[p_, e_] := ((e + 1)*p^(e + 2) - (e + 2)*p^(e + 1) + 1)/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 10 2022 *)

j=[]; for(n=1,200,j=concat(j,sumdiv(n,d,n/d*sigma(d)))); j

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p*X)^2)[n]) /* Ralf Stephan */

N=66; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j); t=1/prod(j=1,N, eta(x^(j)));
t=log(t);t=serconvol(t,c);
Vec(t) /* Joerg Arndt, May 03 2008 */

{ for (n=1, 1000, write("b060640.txt", n, " ", direuler(p=2, n, 1/(1 - X)/(1 - p*X)^2)[n]); ) } /* Harry J. Smith, Jul 08 2009 */

def A060640(n) :
    sigma = sloane.A000203
    return add(sigma(k)*(n/k) for k in divisors(n))
[A060640(i) for i in (1..59)] # Peter Luschny, Sep 15 2012

a(n) = Sum_{d|n} d*tau(d), where tau(d) is the number of divisors of d, cf. A000005. a(n) = Sum_{d|n} d*sigma(n/d), where sigma(n)=sum of divisors of n, cf. A000203. - Vladeta Jovovic, Apr 23 2001
Multiplicative with a(p^e) = ((e+1)*p^{e+2} - (e+2)*p^{e+1} + 1) / (p-1)^2. Dirichlet g.f.: zeta(s)*zeta(s-1)^2. - Franklin T. Adams-Watters, Aug 03 2006
L.g.f.: Sum(A060640(n)*x^n/n) = -log( Product_{j>=1} P(x^j) ) where P(x) = Product_{k>=1} (1-x^k). - Joerg Arndt, May 03 2008
G.f.: Sum_{k>=1} k*tau(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 06 2018
Sum_{k=1..n} a(k) ~ n^2/24 * ((4*gamma - 1)*Pi^2 + 2*Pi^2 * log(n) + 12*Zeta'(2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 01 2019

More terms from James Sellers, Vladeta Jovovic and Matthew Conroy, Apr 17 2001

A123229 Triangle read by rows: T(n, m) = n - (n mod m).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 4, 3, 4, 5, 4, 3, 4, 5, 6, 6, 6, 4, 5, 6, 7, 6, 6, 4, 5, 6, 7, 8, 8, 6, 8, 5, 6, 7, 8, 9, 8, 9, 8, 5, 6, 7, 8, 9, 10, 10, 9, 8, 10, 6, 7, 8, 9, 10, 11, 10, 9, 8, 10, 6, 7, 8, 9, 10, 11, 12, 12, 12, 12, 10, 12, 7, 8, 9, 10, 11, 12, 13, 12, 12, 12, 10, 12, 7, 8, 9, 10, 11, 12, 13

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 06 2006

An equivalent definition: Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127093 from the right. That is, T(n,m) = Sum_{j=m..n} A000012(n,j)*A127093(j,m) = Sum_{j=m..n} A127093(j,m) = m*floor(n/m) = m*A010766(n,m). - Gary W. Adamson, Jan 05 2007

The number of parts k in the triangle is A000203(k) hence the sum of parts k is A064987(k). - Omar E. Pol, Jul 05 2014

			Triangle begins:
{1},
{2, 2},
{3, 2, 3},
{4, 4, 3, 4},
{5, 4, 3, 4, 5},
{6, 6, 6, 4, 5, 6},
{7, 6, 6, 4, 5, 6, 7},
{8, 8, 6, 8, 5, 6, 7, 8},
{9, 8, 9, 8, 5, 6, 7, 8, 9},
...

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A074025, A093960.

Cf. also A024916 (row sums), A127093, A127094, A127096, A127097, A127098, A127099, A038040, A000203, A126988, A127013, A127057.

Flat(List([1..10],n->List([1..n],m->n-(n mod m)))); # Muniru A Asiru, Oct 12 2018

seq(seq(n-modp(n,m),m=1..n),n=1..13); # Muniru A Asiru, Oct 12 2018

a = Table[Table[n - Mod[n, m], {m, 1, n}], {n, 1, 20}]; Flatten[a]

for(n=1,9,for(m=1,n,print1(n-n%m", "))) \\ Charles R Greathouse IV, Nov 07 2011

Edited by N. J. A. Sloane, Jul 05 2014 at the suggestion of Omar E. Pol, who observed that A127095 (Gary W. Adamson, with edits by R. J. Mathar) was the same as this sequence.

cp's OEIS Frontend

A127108 Triangle read by rows, A127099 * A000012.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A127093 Triangle read by rows: T(n,k)=k if k is a divisor of n; otherwise, T(n,k)=0 (1 <= k <= n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Excel

Haskell

Maple

Mathematica

PARI

Formula

A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2*p_i + 3*p_i^2 + ... + (e_i+1)*p_i^e_i).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

PARI

Sage

Formula

Extensions

A123229 Triangle read by rows: T(n, m) = n - (n mod m).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Extensions

A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2p_i + 3p_i^2 + ... + (e_i+1)*p_i^e_i).