A328362 -id:A328362 - cp's OEIS frontend

A245579 Number of odd divisors of n multiplied by n.

1, 2, 6, 4, 10, 12, 14, 8, 27, 20, 22, 24, 26, 28, 60, 16, 34, 54, 38, 40, 84, 44, 46, 48, 75, 52, 108, 56, 58, 120, 62, 32, 132, 68, 140, 108, 74, 76, 156, 80, 82, 168, 86, 88, 270, 92, 94, 96, 147, 150, 204, 104, 106, 216, 220, 112, 228, 116, 118, 240, 122

Offset: 1

Michael Somos, Jul 26 2014

			G.f. = x + 2*x^2 + 6*x^3 + 4*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 8*x^8 + ...
For n = 10 there are two odd divisors of 10: 1 and 5, so a(10) = 2*10 = 20.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
Omar E. Pol, Comments on A245579.

Cf. A000005, A001227, A001511, A001620, A038040, A285891, A299765, A328362, A328365, A352257, A352505.

seq(n*numtheory:-tau(n/2^padic:-ordp(n,2)), n=1..100); # Robert Israel, Apr 26 2017

a[ n_] := If[ n < 1, 0, n Sum[ Mod[d, 2], {d, Divisors @ n}]];
(* Second program: *)
Table[n DivisorSum[n, 1 &, OddQ], {n, 61}] (* Michael De Vlieger, Apr 24 2017 *)

{a(n) = if( n<1, 0, n * sumdiv(n, d, d%2))};

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, if( k%2, k * x^k / (1 - x^k)^2), x * O(x^n)), n))};

{a(n) = if( n<1, 0, n * numdiv(n / 2^valuation(n, 2)))} \\ Fast when n has many divisors. Jens Kruse Andersen, Jul 26 2014

from sympy import divisors
def a(n): return n*len(list(filter(lambda i: i%2==1, divisors(n)))) # Indranil Ghosh, Apr 24 2017

from math import prod
from sympy import factorint
def A245579(n): return n*prod(e+1 for e in factorint(n>>(~n&n-1).bit_length()).values()) # Chai Wah Wu, Dec 31 2023

a(n) is multiplicative with a(2^e) = 2^e, a(p^e) = p^e * (e+1) if p>2.
a(n) = n * A001227(n).
G.f.: Sum_{k>0 odd} k * x^k / (1 - x^k)^2.
From Amiram Eldar, Dec 31 2022: (Start)
Dirichlet g.f.: zeta(s-1)^2*(1-1/2^(s-1)).
Sum_{k=1..n} a(k) ~ n^2*log(n)/4 + (4*gamma + 2*log(2) - 1)*n^2/8, where gamma is Euler's constant (A001620). (End)

Edited by N. J. A. Sloane, Apr 27 2022

A328365 Irregular triangle read by rows, T(n,k), n >= 1, k >= 1, in which row n lists in reverse order the partitions of n into consecutive parts.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 2, 3, 5, 1, 2, 3, 6, 3, 4, 7, 8, 2, 3, 4, 4, 5, 9, 1, 2, 3, 4, 10, 5, 6, 11, 3, 4, 5, 12, 6, 7, 13, 2, 3, 4, 5, 14, 1, 2, 3, 4, 5, 4, 5, 6, 7, 8, 15, 16, 8, 9, 17, 3, 4, 5, 6, 5, 6, 7, 18, 9, 10, 19, 2, 3, 4, 5, 6, 20, 1, 2, 3, 4, 5, 6, 6, 7, 8, 10, 11, 21, 4, 5, 6, 7, 22, 11, 12, 23, 7, 8, 9, 24

Offset: 1

Omar E. Pol, Oct 22 2019

For m >= 0, row 2^m consists of just one element (2^m). - Paolo Xausa, May 24 2025

			Triangle begins:
  [1];
  [2];
  [1, 2], [3];
  [4];
  [2, 3], [5];
  [1, 2, 3], [6];
  [3, 4], [7];
  [8];
  [2, 3, 4], [4, 5], [9];
  [1, 2, 3, 4], [10];
  [5, 6], [11];
  [3, 4, 5], [12];
  [6, 7], [13];
  [2, 3, 4, 5], [14];
  [1, 2, 3, 4, 5], [4, 5, 6], [7, 8], [15];
  [16];
  [8, 9], [17];
  [3, 4, 5, 6], [5, 6, 7], [18];
  [9, 10], [19];
  [2, 3, 4, 5, 6], [20];
  [1, 2, 3, 4, 5, 6], [6, 7, 8], [10, 11], [21];
  [4, 5, 6, 7], [22];
  [11, 12], [23];
  [7, 8, 9], [24];
  [3, 4, 5, 6, 7], [12, 13], [25];
  [5, 6, 7, 8], [26];
  [2, 3, 4, 5, 6, 7], [8, 9, 10], [13, 14], [27];
  [1, 2, 3, 4, 5, 6, 7], [28];
  ...
For n = 9 there are three partitions of 9 into consecutive parts, they are [9], [5, 4], [4, 3, 2], so the 9th row of triangle is [2, 3, 4], [4, 5], [9].
Note that in the below diagram the number of horizontal line segments in the n-th row equals A001227(n), the number of partitions of n into consecutive parts, so we can find the partitions of n into consecutive parts as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [1, 2, 3, 4, 5], [4, 5, 6], [7, 8], [15], equaling the 15th row of the above triangle.
Row        _
  1       |1|_
  2       |_ 2|_
  3       |1|  3|_
  4       |2|_   4|_
  5       |_ 2|    5|_
  6       |1|3|_     6|_
  7       |2|  3|      7|_
  8       |3|_ 4|_       8|_
  9       |_ 2|  4|        9|_
  10      |1|3|  5|_        10|_
  11      |2|4|_   5|         11|_
  12      |3|  3|  6|_          12|_
  13      |4|_ 4|    6|           13|_
  14      |_ 2|5|_   7|_            14|_
  15      |1|3|  4|    7|             15|_
  16      |2|4|  5|    8|_              16|_
  17      |3|5|_ 6|_     8|               17|_
  18      |4|  3|  5|    9|_                18|_
  19      |5|_ 4|  6|      9|                 19|_
  20      |_ 2|5|  7|_    10|_                  20|_
  21      |1|3|6|_   6|     10|                   21|_
  22      |2|4|  4|  7|     11|_                    22|_
  23      |3|5|  5|  8|_      11|                     23|_
  24      |4|6|_ 6|    7|     12|_                      24|_
  25      |5|  3|7|_   8|       12|                       25|_
  26      |6|_ 4|  5|  9|_      13|_                        26|_
  27      |_ 2|5|  6|    8|       13|                         27|_
  28      |1|3|6|  7|    9|       14|                           28|
  ...
The diagram is infinite. For more information about the diagram see A286001.
For an amazing connection with sum of divisors function (A000203) see A237593.

Paolo Xausa, Table of n, a(n) for n = 1..10350 (rows 1..500 of triangle, flattened)

Mirror of A299765.
Row n has length A204217(n).
Row sums give A245579.
Column 1 gives A118235.
Right border gives A000027.
Records give A000027.
Where records occur gives A285899.
The number of partitions into consecutive parts in row n is A001227(n).
For tables of partitions into consecutive parts see A286000 and A286001.
Cf. A000203, A026792, A235791, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774, A328361, A328362.

Table[With[{h = Floor[n/2] - Boole[EvenQ@ n]},Append[Array[Which[Total@ # == n, #, Total@ Most@ # == n, Most[#], True, Nothing] &@ NestWhile[Append[#, #[[-1]] + 1] &, {#}, Total@ # <= n &, 1, h - # + 1] &, h], {n}]], {n, 24}] // Flatten (* Michael De Vlieger, Oct 22 2019 *)

A328361 Triangle read by rows: T(n,k) is the total number of k's in all partitions of n into consecutive parts, (1 <= k <= n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Oct 20 2019

Iff n is a power of 2 (A000079) then row n lists n - 1 zeros together with 1.

Iff n is an odd prime (A065091) then row n lists (n - 3)/2 zeros, 1, 1, (n - 3)/2 zeros, 1.

			Triangle begins:
1;
0, 1;
1, 1, 1;
0, 0, 0, 1;
0, 1, 1, 0, 1;
1, 1, 1, 0, 0, 1;
0, 0, 1, 1, 0, 0, 1;
0, 0, 0, 0, 0, 0, 0, 1;
0, 1, 1, 2, 1, 0, 0, 0, 1;
1, 1, 1, 1, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1;
0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1;
0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
...
For n = 9 there are three partitions of 9 into consecutive parts, they are [9], [5, 4], [4, 3, 2], so the 9th row of triangle is [0, 1, 1, 2, 1, 0, 0, 0, 1].

Row sums give A204217.
Column 1 gives A010054, n >= 1.
Leading diagonal gives A000012.
Cf. A000079, A001227, A065091, A066633, A237048, A237593, A245579, A266531, A285898, A285899, A285900, A285914, A286000, A286001, A299765, A328362.

A285899 Total number of parts in all partitions of all positive integers <= n into consecutive parts.

Original entry on oeis.org

1, 2, 5, 6, 9, 13, 16, 17, 23, 28, 31, 35, 38, 43, 54, 55, 58, 66, 69, 75, 87, 92, 95, 99, 107, 112, 124, 132, 135, 148, 151, 152, 164, 169, 184, 196, 199, 204, 216, 222, 225, 240, 243, 252, 278, 283, 286, 290, 300, 310, 322, 331, 334, 351, 369, 377, 389, 394, 397, 414, 417, 422, 450, 451, 469, 488, 491, 500, 512, 529

Offset: 1

Omar E. Pol, May 02 2017

nonn

Partial sums of A204217.
Sum of first n rows of the triangle A285914.
Where records occur in A328365. - Omar E. Pol, Oct 22 2019
Row sums of A328368. - Omar E. Pol, Nov 04 2019

			For n = 15 there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. The total number of parts in these four partitions is 11, and a(14) = 43, so a(15) = 43 + 11 = 54.

Cf. A001227, A196020, A204217, A235791, A237048, A237591, A237593, A245092, A285900, A285914, A299765, A328361, A328362, A328365, A328368.

A127446 Triangle T(n,k) = n*A051731(n,k) read by rows.

Original entry on oeis.org

1, 2, 2, 3, 0, 3, 4, 4, 0, 4, 5, 0, 0, 0, 5, 6, 6, 6, 0, 0, 6, 7, 0, 0, 0, 0, 0, 7, 8, 8, 0, 8, 0, 0, 0, 8, 9, 0, 9, 0, 0, 0, 0, 0, 9, 10, 10, 0, 0, 10, 0, 0, 0, 0, 10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 12, 12, 12, 12, 0, 12, 0, 0, 0, 0, 0, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 14, 14, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, Jan 14 2007

Replace the 1's in row n of A051731 with n's.

T(n,k) is the sum of the k's in the partitions of n into equal parts. - Omar E. Pol, Nov 25 2019

			First few rows of the triangle:
  1;
  2, 2;
  3, 0, 3;
  4, 4, 0, 4;
  5, 0, 0, 0, 5;
  6, 6, 6, 0, 0, 6;
  7, 0, 0, 0, 0, 0, 7;
  ...
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the sum of the k's are [6, 6, 6, 0, 0, 6] respectively, equaling the 6th row of triangle. - _Omar E. Pol_, Nov 25 2019

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A038040 (row sums), A051731, A126988, A244051, A328362.

a127446 n k = a127446_tabl !! (n-1) !! (k-1)
a127446_row n = a127446_tabl !! (n-1)
a127446_tabl = zipWith (\v ws -> map (* v) ws) [1..] a051731_tabl
-- Reinhard Zumkeller, Jan 21 2014

A127446 := proc(n,k) if n mod k = 0 then n; else 0; fi; end: for n from 1 to 20 do for k from 1 to n do printf("%d,",A127446(n,k)) ; od: od: # R. J. Mathar, May 08 2009

Flatten[Table[If[Mod[n, k] == 0, n, 0], {n, 20}, {k, n}]] (* Vincenzo Librandi, Nov 02 2016 *)

T(n,k) = k*A126988(n,k). - Omar E. Pol, Nov 25 2019

Edited and extended by R. J. Mathar, May 08 2009

A328371 Irregular triangle read by rows: T(n,k) is the sum of all parts of all partitions of all positive integers <= n into k consecutive parts.

Original entry on oeis.org

1, 3, 6, 3, 10, 3, 15, 8, 21, 8, 6, 28, 15, 6, 36, 15, 6, 45, 24, 15, 55, 24, 15, 10, 66, 35, 15, 10, 78, 35, 27, 10, 91, 48, 27, 10, 105, 48, 27, 24, 120, 63, 42, 24, 15, 136, 63, 42, 24, 15, 153, 80, 42, 24, 15, 171, 80, 60, 42, 15, 190, 99, 60, 42, 15, 210, 99, 60, 42, 35, 231, 120, 81, 42, 35, 21

Offset: 1

Omar E. Pol, Nov 02 2019

Column k lists the partial sums of the k-th column of triangle A285891.

			Triangle begins:
    1;
    3;
    6,   3;
   10,   3;
   15,   8;
   21,   8,   6;
   28,  15,   6;
   36,  15,   6;
   45,  24,  15;
   55,  24,  15, 10;
   66,  35,  15, 10;
   78,  35,  27, 10;
   91,  48,  27, 10;
  105,  48,  27, 24,
  120,  63,  42, 24, 15;
  136,  63,  42, 24, 15;
  153,  80,  42, 24, 15;
  171,  80,  60, 42, 15;
  190,  99,  60, 42, 15;
  210,  99,  60, 42, 35;
  231, 120,  81, 42, 35, 21;
  253, 120,  81, 64, 35, 21;
  276, 143,  81, 64, 35, 21;
  300, 143, 105, 64, 35, 21;
  325, 168, 105, 64, 60, 21;
  351, 168, 105, 90, 60, 21;
  378, 195, 132, 90, 60, 48;
  406, 195, 132, 90, 60, 48, 28;
...

Row sums give A285900.
Row n has length A003056(n).
Column 1 gives the nonzero terms of A000217.
Column k starts with A000217(k) in the row A000217(k).
Cf. A196020, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A245579, A262612, A285899, A285914, A285891, A286000, A286001, A286013, A299765, A328362, A328365, A328368.

tt(n, k) = n*(if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0)); \\ A285891
t(n, k) = sum(j=k*(k+1)/2, n, tt(j, k));
tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 04 2019

cp's OEIS Frontend

A245579 Number of odd divisors of n multiplied by n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Formula

Extensions

A328365 Irregular triangle read by rows, T(n,k), n >= 1, k >= 1, in which row n lists in reverse order the partitions of n into consecutive parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A328361 Triangle read by rows: T(n,k) is the total number of k's in all partitions of n into consecutive parts, (1 <= k <= n).

Views

Author

Keywords

Comments

Examples

Crossrefs

A285899 Total number of parts in all partitions of all positive integers <= n into consecutive parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

A127446 Triangle T(n,k) = n*A051731(n,k) read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A328371 Irregular triangle read by rows: T(n,k) is the sum of all parts of all partitions of all positive integers <= n into k consecutive parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI