A126988 -id:A126988 - cp's OEIS frontend

A026741 a(n) = n if n odd, n/2 if n even.

0, 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 16, 33, 17, 35, 18, 37, 19, 39, 20, 41, 21, 43, 22, 45, 23, 47, 24, 49, 25, 51, 26, 53, 27, 55, 28, 57, 29, 59, 30, 61, 31, 63, 32, 65, 33, 67, 34, 69, 35, 71, 36, 73, 37, 75, 38

Offset: 0

J. Carl Bellinger (carlb(AT)ctron.com)

a(n) is the size of largest conjugacy class in D_2n, the dihedral group with 2n elements. - Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002
a(n+1) is the composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type D_4 (quaternion group). - Paul Boddington, Oct 23 2003
For n > 1, a(n) is the greatest common divisor of all permutations of {0, 1, ..., n} treated as base n + 1 integers. - David Scambler, Nov 08 2006 (see the Mathematics Stack Exchange link below).
From Dimitrios Choussos (choussos(AT)yahoo.de), May 11 2009: (Start)
Sequence A075888 and the above sequence are fitting together.
First 2 entries of this sequence have to be taken out.
In some cases two three or more sequenced entries of this sequence have to be added together to get the next entry of A075888.
Example: Sequences begin with 1, 3, 2, 5, 3, 7, 4, 9 (4 + 9 = 13, the next entry in A075888).
But it works out well up to primes around 50000 (haven't tested higher ones).
As A075888 gives a very regular graph. There seems to be a regularity in the primes. (End)
Starting with 1 = triangle A115359 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 27 2009
From Gary W. Adamson, Dec 11 2009: (Start)
Let M be an infinite lower triangular matrix with (1, 1, 1, 0, 0, 0, ...) in every column, shifted down twice. This sequence starting with 1 = M * (1, 2, 3, ...)
M =
  1;
  1, 0;
  1, 1, 0;
  0, 1, 0, 0;
  0, 1, 1, 0, 0;
  0, 0, 1, 0, 0, 0;
  0, 0, 1, 1, 0, 0, 0;
  ...
A026741 = M * (1, 2, 3, ...); but A002487 = lim_{n->infinity} M^n, a left-shifted vector considered as a sequence. (End)
A particular case of sequence for which a(n+3) = (a(n+2) * a(n+1)+q)/a(n) for every n > n0. Here n0 = 1 and q = -1. - Richard Choulet, Mar 01 2010
For n >= 2, a(n+1) is the smallest m such that s_n(2*m*(n-1))/(n-1) is even, where s_b(c) is the sum of digits of c in base b. - Vladimir Shevelev, May 02 2011
A001477 and A005408 interleaved. - Omar E. Pol, Aug 22 2011
Numerator of n/((n-1)*(n-2)). - Michael B. Porter, Mar 18 2012
Number of odd terms of n-th row in the triangles A162610 and A209297. - Reinhard Zumkeller, Jan 19 2013
For n >= 3, a(n) is the periodic of integer of spiral length ratio of spiral that have (n-1) circle centers. See illustration in links. - Kival Ngaokrajang, Dec 28 2013
This is the sequence of Lehmer numbers u_n(sqrt(R), Q) with the parameters R = 4 and Q = 1. It is a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all natural numbers n and m. Cf. A005013 and A108412. - Peter Bala, Apr 18 2014
The sequence of convergents of the 2-periodic continued fraction [0; 1, -4, 1, -4, ...] = 1/(1 - 1/(4 - 1/(1 - 1/(4 - ...)))) = 2 begins [0/1, 1/1, 4/3, 3/2, 8/5, 5/3, 12/7, ...]. The present sequence is the sequence of denominators; the sequence of numerators of the continued fraction convergents [0, 1, 4, 3, 8, 5, 12, ...] is A022998, also a strong divisibility sequence. - Peter Bala, May 19 2014
For n >= 3, (a(n-2)/a(n))*Pi = vertex angle of a regular n-gon. See illustration in links. - Kival Ngaokrajang, Jul 17 2014
For n > 1, the numerator of the harmonic mean of the first n triangular numbers. - Colin Barker, Nov 13 2014
The difference sequence is a permutation of the integers. - Clark Kimberling, Apr 19 2015
From Timothy Hopper, Feb 26 2017: (Start)
Given the function a(n, p) = n/p if n mod p = 0, else n, then a possible formula is: a(n, p) = n*(1 + (p-1)*((n^(p-1)) mod p))/p, p prime, (n^(p-1)) mod p = 1, n not divisible by p. (Fermat's Little Theorem). Examples: p = 2; a(n), p = 3; A051176(n), p = 5; A060791(n), p = 7; A106608(n).
Conjecture: lcm(n, p) = p*a(n, p), gcd(n, p) = n/a(n, p). (End)
Let r(n) = (a(n+1) + 1)/a(n+1) if n mod 2 = 1, a(n+1)/(a(n+1) + 2) otherwise; then lim_{k->oo} 2^(k+2) * Product_{n=0..k} r(n)^(k-n) = Pi. - Dimitris Valianatos, Mar 22 2021
Number of integers k from 1 to n such that gcd(n,k) is odd. - Amiram Eldar, May 18 2025

			G.f. = x + x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 3*x^6 + 7*x^7 + 4*x^8 + ...

David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005), p. 53.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, 2nd Ed. Penguin (1997), p. 79.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv:1105.3399 [math.GM], 2011.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2.
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Y. Ito and I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999.
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.
Mathematics Stack Exchange, Permutations (with no duplicates) of decimal base digits 1,2,...,8,9,0.
Kival Ngaokrajang, Illustration of spiral with circle centers 2..5.
Kival Ngaokrajang, Illustration of vertex angle of regular n-gon for n = 3..7.
Eric Weisstein's World of Mathematics, Lehmer Number.
Eric Weisstein's World of Mathematics, Simplex Simplex Picking.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
Index to divisibility sequences.

Signed version is in A030640. Partial sums give A001318.
Cf. A051176, A060819, A060791, A060789 for n / gcd(n, k) with k = 3..6. See also A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).
Cf. A045896, A022998, A060762, A126988, A109007, A130334, A109043, A115359, A002487, A220466.
Cf. A013942.
Cf. A227042 (first column). Cf. A005013 and A108412.
Cf. A000217, A256095.

import Data.List (transpose)
a026741 n = a026741_list !! n
a026741_list = concat $ transpose [[0..], [1,3..]]
-- Reinhard Zumkeller, Dec 12 2011

[2*n/(3+(-1)^n): n in [0..70]]; // Vincenzo Librandi, Aug 14 2011

A026741 := proc(n) if type(n,'odd') then n; else n/2; end if; end proc: seq(A026741(n), n=0..76); # R. J. Mathar, Jan 22 2011

Numerator[Abs[Table[Det[DiagonalMatrix[Table[1/i^2 - 1, {i, 1, n - 1}]] + 1], {n, 20}]]] (* Alexander Adamchuk, Jun 02 2006 *)
halfMax = 40; Riffle[Range[0, halfMax], Range[1, 2halfMax + 1, 2]] (* Harvey P. Dale, Mar 27 2011 *)
a[ n_] := Numerator[n / 2]; (* Michael Somos, Jan 20 2017 *)
Array[If[EvenQ[#],#/2,#]&,80,0] (* Harvey P. Dale, Jul 08 2023 *)

a(n) = numerator(n/2) \\ Rick L. Shepherd, Sep 12 2007

def A026741(n): return n if n % 2 else n//2 # Chai Wah Wu, Apr 02 2021

[lcm(n, 2) / 2 for n in range(77)] # Zerinvary Lajos, Jun 07 2009

G.f.: x*(1 + x + x^2)/(1-x^2)^2. - Len Smiley, Apr 30 2001
a(n) = 2*a(n-2) - a*(n-4) for n >= 4.
a(n) = n * 2^((n mod 2) - 1). - Reinhard Zumkeller, Oct 16 2001
a(n) = 2*n/(3 + (-1)^n). - Benoit Cloitre, Mar 24 2002
Multiplicative with a(2^e) = 2^(e-1) and a(p^e) = p^e, p > 2. - Vladeta Jovovic, Apr 05 2002
a(n) = n / gcd(n, 2). a(n)/A045896(n) = n/((n+1)*(n+2)).
For n > 0, a(n) = denominator of Sum_{i=1..n-1} 2/(i*(i+1)), numerator=A022998. - Reinhard Zumkeller, Apr 21 2012, Jul 25 2002 [thanks to Phil Carmody who noticed an error]
For n > 1, a(n) = GCD of the n-th and (n-1)-th triangular numbers (A000217). - Ross La Haye, Sep 13 2003
Euler transform of finite sequence [1, 2, -1]. - Michael Somos, Jun 15 2005
G.f.: x * (1 - x^3) / ((1 - x) * (1 - x^2)^2) = Sum_{k>0} k * (x^k - x^(2*k)). - Michael Somos, Jun 15 2005
a(n+3) + a(n+2) = 3 + a(n+1) + a(n). a(n+3) * a(n) = - 1 + a(n+2) * a(n+1). a(n) = -a(-n) for all n in Z. - Michael Somos, Jun 15 2005
For n > 1, a(n) is the numerator of the average of 1, 2, ..., n - 1; i.e., numerator of A000217(n-1)/(n-1), with corresponding denominators [1, 2, 1, 2, ...] (A000034). - Rick L. Shepherd, Jun 05 2006
Equals A126988 * (1, -1, 0, 0, 0, ...). - Gary W. Adamson, Apr 17 2007
For n >= 1, a(n) = gcd(n,A000217(n)). - Rick L. Shepherd, Sep 12 2007
a(n) = numerator(n/(2*n-2)) for n >= 2; A022998(n-1) = denominator(n/(2*n-2)) for n >= 2. - Johannes W. Meijer, Jun 18 2009
a(n) = A167192(n+2, 2). - Reinhard Zumkeller, Oct 30 2009
a(n) = A106619(n) * A109012(n). - Paul Curtz, Apr 04 2011
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109043(n)/2.
Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s). (End)
a(n) = A001318(n) - A001318(n-1) for n > 0. - Jonathan Sondow, Jan 28 2013
a((2*n+1)*2^p - 1) = 2^p - 1 + n*A151821(p+1), p >= 0 and n >= 0. - Johannes W. Meijer, Feb 03 2013
a(n+1) = denominator(H(n, 1)), n >= 0, with H(n, 1) = 2*n/(n+1) the harmonic mean of n and 1. a(n+1) = A227042(n, 1). See the formula a(n) = n/gcd(n, 2) given above. - Wolfdieter Lang, Jul 04 2013
a(n) = numerator(n/2). - Wesley Ivan Hurt, Oct 02 2013
a(n) = numerator(1 - 2/(n+2)), n >= 0; a(n) = denominator(1 - 2/n), n >= 1. - Kival Ngaokrajang, Jul 17 2014
a(n) = Sum_{i = floor(n/2)..floor((n+1)/2)} i. - Wesley Ivan Hurt, Apr 27 2016
Euler transform of length 3 sequence [1, 2, -1]. - Michael Somos, Jan 20 2017
G.f.: x / (1 - x / (1 - 2*x / (1 + 7*x / (2 - 9*x / (7 - 4*x / (3 - 7*x / (2 + 3*x))))))). - Michael Somos, Jan 20 2017
From Peter Bala, Mar 24 2019: (Start)
a(n) = Sum_{d|n, n/d odd} phi(d), where phi(n) is the Euler totient function A000010.
O.g.f.: Sum_{n >= 1} phi(n)*x^n/(1 - x^(2*n)). (End)
a(n) = A256095(2*n,n). - Alois P. Heinz, Jan 21 2020
E.g.f.: x*(2*cosh(x) + sinh(x))/2. - Stefano Spezia, Apr 28 2023
From Ctibor O. Zizka, Oct 05 2023: (Start)
For k >= 0, a(k) = gcd(k + 1, k*(k + 1)/2).
If (k mod 4) = 0 or 2 then a(k) = (k + 1).
If (k mod 4) = 1 or 3 then a(k) = (k + 1)/2. (End)
Sum_{n=1..oo} 1/a(n)^2 = 7*Pi^2/24. - Stefano Spezia, Dec 02 2023
a(n)*a(n+1) = A000217(n). - Rémy Sigrist, Mar 19 2025

Better description from Jud McCranie

Edited by Ralf Stephan, Jun 04 2003

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

A054523 Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 4, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 6, 0, 2, 0, 0, 0, 0, 0, 1, 4, 4, 0, 0, 1, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 6

Offset: 1

N. J. A. Sloane, Apr 09 2000

From Gary W. Adamson, Jan 08 2007: (Start)
Let H be this lower triangular matrix. Then:
H * A051731 = A126988,
H * [1, 2, 3, ...] = 1, 3, 5, 8, 9, 15, ... = A018804,
H * sigma(n) = A038040 = d(n) * n = 1, 4, 6, 12, 10, ... where sigma(n) = A000203,
H * d(n) (A000005) = sigma(n) = A000203,
Row sums are A000027 (corrected by Werner Schulte, Sep 06 2020, see comment of Gary W. Adamson, Aug 03 2008),
H^2 * d(n) = d(n)*n, H^2 = A127192,
H * mu(n) (A008683) = A007431(n) (corrected by Werner Schulte, Sep 06 2020),
H^2 row sums = A018804. (End)
The Möbius inversion principle of Richard Dedekind and Joseph Liouville (1857), cf. "Concrete Mathematics", p. 136, is equivalent to the statement that row sums are the row index n. - Gary W. Adamson, Aug 03 2008
The multivariable row polynomials give n times the cycle index for the cyclic group C_n, called Z(C_n) (see the MathWorld link with the Harary reference): n*Z(C_n) = Sum_{k=1..n} T(n,k)*(y_{n/k})^k, n >= 1. E.g., 6*Z(C_6) = 2*(y_6)^1 + 2*(y_3)^2 + 1*(y_2)^3 + 1*(y_1)^6. - Wolfdieter Lang, May 22 2012
See A102190 (no 0's, rows reversed). - Wolfdieter Lang, May 29 2012
This is the number of permutations in the n-th cyclic group which are the product of k disjoint cycles. - Robert A. Beeler, Aug 09 2013

			Triangle begins
   1;
   1, 1;
   2, 0, 1;
   2, 1, 0, 1;
   4, 0, 0, 0, 1;
   2, 2, 1, 0, 0, 1;
   6, 0, 0, 0, 0, 0, 1;
   4, 2, 0, 1, 0, 0, 0, 1;
   6, 0, 2, 0, 0, 0, 0, 0, 1;
   4, 4, 0, 0, 1, 0, 0, 0, 0, 1;
  10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1;

Ronald L. Graham, D. E. Knuth, Oren Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994, p. 136.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Cycle Index.

Cf. A000005, A000007, A000010, A000012, A000203, A007431, A008683.
Cf. A038040, A051731, A054521, A054525, A102190, A126988.
Sums incliude: A029935, A069097, A092843 (diagonal), A209295.
Sums of the form Sum_{k} k^p * T(n, k): A000027 (p=0), A018804 (p=1), A069097 (p=2), A343497 (p=3), A343498 (p=4), A343499 (p=5).

a054523 n k = a054523_tabl !! (n-1) !! (k-1)
a054523_row n = a054523_tabl !! (n-1)
a054523_tabl = map (map (\x -> if x == 0 then 0 else a000010 x)) a126988_tabl
-- Reinhard Zumkeller, Jan 20 2014

A054523:= func< n,k | k eq n select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
[A054523(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 24 2024

A054523 := proc(n,k) if n mod k = 0 then numtheory[phi](n/k) ; else 0; end if; end proc: # R. J. Mathar, Apr 11 2011

T[n_, k_]:= If[k==n,1,If[Divisible[n, k], EulerPhi[n/k], 0]];
Table[T[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Dec 15 2017 *)

for(n=1, 10, for(k=1, n, print1(if(!(n % k), eulerphi(n/k), 0), ", "))) \\ G. C. Greubel, Dec 15 2017

def A054523(n,k):
    if (k==n): return 1
    elif (n%k)==0: return euler_phi(int(n//k))
    else: return 0
flatten([[A054523(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 24 2024

Sum_{k=1..n} k * T(n, k) = A018804(n). - Gary W. Adamson, Jan 08 2007
Equals A054525 * A126988 as infinite lower triangular matrices. - Gary W. Adamson, Aug 03 2008
From Werner Schulte, Sep 06 2020: (Start)
Sum_{k=1..n} T(n,k) * A000010(k) = A029935(n) for n > 0.
Sum_{k=1..n} k^2 * T(n,k) = A069097(n) for n > 0. (End)
From G. C. Greubel, Jun 24 2024: (Start)
T(2*n-1, n) = A000007(n-1), n >= 1.
T(2*n, n) = A000012(n), n >= 1.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1 - (-1)^n)*n/2.
Sum_{k=1..floor(n+1)/2} T(n-k+1, k) = A092843(n+1).
Sum_{k=1..n} (k+1)*T(n, k) = A209295(n).
Sum_{k=1..n} k^3 * T(n, k) = A343497(n).
Sum_{k=1..n} k^4 * T(n, k) = A343498(n).
Sum_{k=1..n} k^5 * T(n, k) = A343499(n). (End)

A080512 a(n) = n if n is odd, a(n) = 3*n/2 if n is even.

Original entry on oeis.org

1, 3, 3, 6, 5, 9, 7, 12, 9, 15, 11, 18, 13, 21, 15, 24, 17, 27, 19, 30, 21, 33, 23, 36, 25, 39, 27, 42, 29, 45, 31, 48, 33, 51, 35, 54, 37, 57, 39, 60, 41, 63, 43, 66, 45, 69, 47, 72, 49, 75, 51, 78, 53, 81, 55, 84, 57, 87, 59, 90, 61, 93, 63, 96, 65, 99, 67, 102

Offset: 1

Amarnath Murthy, Mar 20 2003

First differences of the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011
Last term in n-th row of A080511.
Also A005408 and positive terms of A008585 interleaved. - Omar E. Pol, May 28 2012
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized heptagonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A080511, A008619, A126988.

Cf. A005408, A008585, A064455, A085787, A126246

import Data.List (transpose)
a080512 n = if m == 0 then 3 * n' else n  where (n', m) = divMod n 2
a080512_list = concat $ transpose [[1, 3 ..], [3, 6 ..]]
-- Reinhard Zumkeller, Apr 06 2015

[n*(5+(-1)^n)/4: n in [1..60]]; // Vincenzo Librandi, Sep 11 2011

Table[If[EvenQ[n],3n/2,n],{n,68}] (* Jayanta Basu, May 20 2013 *)

a(n) = n if n is odd, a(n) = 3*n/2 if n is even.
a(n)*a(n+3) = -3 + a(n+1)*a(n+2).
From Paul Barry, Sep 04 2003: (Start)
G.f.: (1+3*x+x^2)/((1-x^2)^2);
a(n) = n*(5 + (-1)^n)/4. (End)
Multiplicative with a(2^e) = 3*2^(e-1), a(p^e) = p^e otherwise. - Christian G. Bower, May 17 2005
Equals A126988 * (1, 1, 0, 0, 0, ...) - Gary W. Adamson, Apr 17 2007
Dirichlet g.f.: zeta(s-1) * (1 + 1/2^s). - Amiram Eldar, Oct 25 2023
Sum_{d divides n} mu(n/d)*a(d) = A126246(n), where mu(n) = A008683(n) is the Möbius function. - Peter Bala, Dec 31 2023

A113704 Triangle read by rows. The indicator function for divisibility.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Nov 05 2005

From Peter Luschny, Jul 01 2023: (Start)
Definition: d divides n <=> n = m*d for some m.
Equivalently, d divides n iff d = n or d > 0, and the integer remainder of n divided by d is 0.
This definition is sufficient to define the infinite lower triangular array, i.e., if we consider only the range 0 <= d <= n. But see the construction of the inverse square array in A363914, which has to make this restriction explicit because with the above definition every integer divides 0, and thus the first row of the square matrix becomes 1 for all d.
(End)

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 1, 0, 1;
  0, 1, 1, 0, 1;
  0, 1, 0, 0, 0, 1;
  0, 1, 1, 1, 0, 0, 1;
  0, 1, 0, 0, 0, 0, 0, 1;

Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.

Wikipedia, Indicator function

Cf. A051731, A113705 (reversed rows concatenated).

Cf. A000005 (row sums), A000007, A000961, A007947, A057427, A126988, A363914 (inverse triangle).

divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):
A113704_row := n -> local k; seq(divides(k, n), k = 0..n):
seq(print(A113704_row(n)), n = 0..9);  # Peter Luschny, Jun 28 2023

Table[If[k==0,Boole[n==0],Boole[Divisible[n,k]]],{n,0,10},{k,0,n}] (* Gus Wiseman, Mar 06 2020 *)

def A113704_row(n): return [int(k.divides(n)) for k in (0..n)]
for n in (0..9): print(A113704_row(n))  # Peter Luschny, Jun 28 2023

dim = 10
matrix(ZZ, dim, dim, lambda n, d: d <= n and ZZ(d).divides(ZZ(n)))  # Peter Luschny, Jul 01 2023

Column k has g.f. 1/(1-x^k), k >= 1. Column 0 has g.f. 1.

T(n, d) = 1 if d > 0 and d|n, otherwise 0^n. - Gus Wiseman, Mar 06 2020

Name edited by Peter Luschny, Jul 29 2023

A244051 Triangle read by rows in which row n lists the parts of the partitions of n into equal parts, in nonincreasing order.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 6, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 9, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 5, 5, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 08 2014

Row n has length sigma(n) = A000203(n).
Row sums give n*A000005(n) = A038040(n).
Column 1 is A000027.
Both columns 2 and 3 are A032742, n > 1.
For any k > 0 and t > 0, the sequence contains exactly one run of k consecutive t's. - Rémy Sigrist, Feb 11 2019
From Omar E. Pol, Dec 04 2019: (Start)
The number of parts congruent to 0 (mod m) in row m*n equals sigma(n) = A000203(n).
The number of parts greater than 1 in row n equals A001065(n), the sum of aliquot parts of n.
The number of parts greater than 1 and less than n in row n equals A048050(n), the sum of divisors of n except for 1 and n.
The number of partitions in row n equals A000005(n), the number of divisors of n.
The number of partitions in row n with an odd number of parts equals A001227(n).
The sum of odd parts in row n equals the sum of parts of the partitions in row n that have an odd number of parts, and equals the sum of all parts in the partitions of n into consecutive parts, and equals A245579(n) = n*A001227(n).
The decreasing records in row n give the n-th row of A056538.
Row n has n 1's which are all at the end of the row.
First n rows contain A000217(n) 1's.
The number of k's in row n is A126988(n,k).
The number of odd parts in row n is A002131(n).
The k-th block in row n has A027750(n,k) parts.
Right border gives A000012. (End)
The r-th row of the triangle begins at index k = A160664(r-1). - Samuel Harkness, Jun 21 2022

			Triangle begins:
   [1];
   [2], [1,1];
   [3], [1,1,1];
   [4], [2,2], [1,1,1,1];
   [5], [1,1,1,1,1];
   [6], [3,3], [2,2,2], [1,1,1,1,1,1];
   [7], [1,1,1,1,1,1,1];
   [8], [4,4], [2,2,2,2], [1,1,1,1,1,1,1,1];
   [9], [3,3,3], [1,1,1,1,1,1,1,1,1];
  [10], [5,5], [2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1];
  [11], [1,1,1,1,1,1,1,1,1,1,1];
  [12], [6,6], [4,4,4], [3,3,3,3], [2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1];
  [13], [1,1,1,1,1,1,1,1,1,1,1,1,1];
  [14], [7,7], [2,2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  [15], [5,5,5], [3,3,3,3,3], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  [16], [8,8], [4,4,4,4], [2,2,2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  ...
For n = 6 the 11 partitions of 6 are [6], [3, 3], [4, 2], [2, 2, 2], [5, 1], [3, 2], [4, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]. There are only four partitions of 6 that contain equal parts so the 6th row of triangle is [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. The number of parts equals sigma(6) = A000203(6) = 12. The row sum is A038040(6) = 6*A000005(6) = 6*4 = 24.
From _Omar E. Pol_, Dec 04 2019: (Start)
The structure of the above triangle is as follows:
   1;
   2 11;
   3    111;
   4 22     1111;
   5             11111;
   6 33 222            111111;
   7                          1111111;
   8 44     2222                      11111111;
   9    333                                    111111111;
  ... (End)

Samuel Harkness, Table of n, a(n) for n = 1..10000
Samuel Harkness, Log-log Scatterplot of the first 1000000 terms

Cf. A000005, A000012, A000041, A000203, A000217, A001065, A001227, A002131, A027750, A032742, A038040, A048050, A056538, A126988, A160664, A237593, A245579, A299765, A328365, A309400 (mirror).

A244051row[n_]:=Flatten[Map[ConstantArray[#,n/#]&,Reverse[Divisors[n]]]];
Array[A244051row,10] (* Paolo Xausa, Oct 16 2023 *)

tabf(nn) = {for (n=1, nn, d = Vecrev(divisors(n)); for (i=1, #d, for (j=1, n/d[i], print1(d[i], ", "));); print(););} \\ Michel Marcus, Nov 08 2014

A038046 Shifts left under transform T where Ta is (identity) DCONV a.

Original entry on oeis.org

1, 1, 3, 6, 12, 17, 32, 39, 63, 81, 120, 131, 213, 226, 311, 377, 503, 520, 742, 761, 1031, 1169, 1442, 1465, 2008, 2093, 2558, 2801, 3465, 3494, 4591, 4622, 5628, 6054, 7111, 7390, 9321, 9358, 10899, 11616, 13873, 13914, 17070, 17113, 20063, 21509, 24462

Offset: 1

Christian G. Bower

Eigensequence of triangle A126988. (i.e. the sequence shifts upon multiplication from the left by triangle A126988). - Gary W. Adamson, Apr 27 2009

Number of planted achiral trees with a distinguished leaf. - Gus Wiseman, Jul 31 2018

			From _Gus Wiseman_, Jul 31 2018: (Start)
The a(5) = 12 planted achiral trees with a distinguished leaf:
  (Oooo), (oOoo), (ooOo), (oooO),
  ((O)(o)), ((o)(O)),
  ((Ooo)), ((oOo)), ((ooO)),
  (((Oo))), (((oO))),
  ((((O)))).
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A126988. - Gary W. Adamson, Apr 27 2009

Cf. A000081, A002033, A003238, A004111, A007554, A038046, A067824, A317580.

a:= proc(n) option remember; `if`(n<2, n, (m-> m*
      add(a(d)/d, d=numtheory[divisors](m)))(n-1))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 09 2019

a[n_]:=If[n==1,1,Sum[d*a[(n-1)/d],{d,Divisors[n-1]}]];
Array[a,30] (* Gus Wiseman, Jul 31 2018 *)

a(1) = 1; a(n > 1) = Sum_{d|(n-1)} d * a((n-1)/d). - Gus Wiseman, Jul 31 2018

G.f. A(x) satisfies: A(x) = x * (1 + Sum_{j>=1} j*A(x^j)). - Ilya Gutkovskiy, May 09 2019

A112624 If p^b(p,n) is the highest power of the prime p dividing n, then a(n) = Product_{p|n} b(p,n)!.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 1, 1, 2, 1, 1, 1, 24, 1, 2, 1, 2, 1, 1, 1, 6, 2, 1, 6, 2, 1, 1, 1, 120, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 24, 2, 2, 1, 2, 1, 6, 1, 6, 1, 1, 1, 2, 1, 1, 2, 720, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 2, 2, 1, 1, 1, 24, 24, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 120, 1, 2, 2, 4, 1

Offset: 1

Leroy Quet, Dec 25 2005

The logarithm of the Dirichlet series with the reciprocals of this sequence as coefficients is the Dirichlet series with the characteristic function of primes A010051 as coefficients. - Mats Granvik, Apr 13 2011

			45 = 3^2 * 5^1. So a(45) = 2! * 1! = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
Index entries for sequences computed from exponents in factorization of n.

For row > 1: a(n) = row products of A100995(A126988), when neglecting zero elements.

Cf. A000012, A000142, A010051, A028234, A067029, A112622, A112623, A156552, A240953, A246660, A306694.

w := n -> op(2, ifactors(n)): a := n -> mul(factorial(w(n)[j][2]), j = 1..nops(w(n))): seq(a(n), n = 1..101); # Emeric Deutsch, May 17 2012

f[n_] := Block[{fi = Last@Transpose@FactorInteger@n}, Times @@ (fi!)]; Array[f, 101] (* Robert G. Wilson v, Dec 27 2005 *)

A112624(n) = { my(f = factor(n), m = 1); for (k=1, #f~, m *= f[k, 2]!; ); m; } \\ Antti Karttunen, May 28 2017

def A112624(n):
    return mul(factorial(s[1]) for s in factor(n))
[A112624(i) for i in (1..101)]  # Peter Luschny, Jun 15 2013

(define (A112624 n) (if (= 1 n) n (* (A000142 (A067029 n)) (A112624 (A028234 n))))) ;; Antti Karttunen, May 29 2017

From Antti Karttunen, May 29 2017: (Start)
a(1) = 1 and for n > 1, a(n) = A000142(A067029(n)) * a(A028234(n)).
a(n) = A246660(A156552(n)). (End)
From Mats Granvik, Mar 05 2019: (Start)
log(a(n)) = inverse Möbius transform of log(A306694(n)).
log(a(n)) = Sum_{k=1..n} [k|n]*log(A306694(n/k))*A000012(k). (End)
From Amiram Eldar, Mar 08 2024: (Start)
Let f(n) = 1/a(n). Formulas from Jakimczuk (2024, pp. 12-15):
Dirichlet g.f. of f(n): Sum_{n>=1} f(n)/n^s = exp(P(s)), where P(s) is the prime zeta function.
Sum_{k=1..n} f(k) = c * n + o(n), where c = A240953.
Sum_{k=1..n} f(k)/k = c * log(n) + o(log(n)), where c = A240953. (End)

More terms from Robert G. Wilson v, Dec 27 2005

A127170 Triangle read by rows: T(n,k) is the number of divisors of n that are divisible by k, with 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 3, 2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 4, 3, 0, 2, 0, 0, 0, 1, 3, 0, 2, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 4, 3, 2, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Column k lists the terms of A000005 interleaved with k - 1 zeros.

Eigensequence of the triangle = A007557; i.e., sequence A007557 shifts to the left upon multiplication by A127170. - Gary W. Adamson, Apr 27 2009

			First 10 rows of the triangle:
  1;
  2, 1;
  2, 0, 1;
  3, 2, 0, 1;
  2, 0, 0, 0, 1;
  4, 2, 2, 0, 0, 1;
  2, 0, 0, 0, 0, 0, 1;
  4, 3, 0, 2, 0, 0, 0, 1;
  3, 0, 2, 0, 0, 0, 0, 0, 1;
  4, 2, 0, 0, 2, 0, 0, 0, 0, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)

Row sums give A007425.

Cf. A000005, A007557, A051731, A126988, A007429, A127170, A195050.

T:= (n, k)-> `if`(irem(n, k)=0, numtheory[tau](n/k), 0):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Feb 16 2022

Table[Function[D, Table[Count[D, ?(Mod[#, k] == 0 &)], {k, n}]]@ Divisors[n], {n, 12}] // Flatten (* _Michael De Vlieger, Feb 16 2022 *)

tabl(nn) = {m = matrix(nn, nn, n, k, if ((n % k) == 0, 1, 0)); m = m^2; for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ Michel Marcus, Apr 01 2015

A007429(n) = Sum_{i=1..n} i*a(i).

T(n,k) = A000005(n/k), if k divides n, otherwise 0, with n >= 1 and 1 <= k <= n. - Omar E. Pol, Apr 01 2015

8 terms taken from Example section and then corrected in Data section by Omar E. Pol, Mar 30 2015
Extended beyond a(21) by Omar E. Pol, Apr 01 2015
New name (which was a comment dated Mar 30 2015) from Omar E. Pol, Feb 16 2022

cp's OEIS Frontend

A026741 a(n) = n if n odd, n/2 if n even.

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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).

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A054523 Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n).

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A080512 a(n) = n if n is odd, a(n) = 3*n/2 if n is even.

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A113704 Triangle read by rows. The indicator function for divisibility.

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