A245579 -id:A245579 - cp's OEIS frontend

A326125 Expansion of Sum_{k>=1} k^2 * x^k / (1 + x^k)^2.

1, 2, 12, 4, 30, 24, 56, 8, 117, 60, 132, 48, 182, 112, 360, 16, 306, 234, 380, 120, 672, 264, 552, 96, 775, 364, 1080, 224, 870, 720, 992, 32, 1584, 612, 1680, 468, 1406, 760, 2184, 240, 1722, 1344, 1892, 528, 3510, 1104, 2256, 192, 2793, 1550, 3672, 728, 2862, 2160

Offset: 1

Ilya Gutkovskiy, Sep 10 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000593, A064987, A078306, A143520, A185152, A245579, A326238, A353908.

nmax = 54; CoefficientList[Series[Sum[k^2 x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[n Sum[(-1)^(n/d + 1) d, {d, Divisors[n]}], {n, 1, 54}]
f[p_, e_] := p^e*(p^(e+1)-1)/(p-1); f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

a(n)={n*sumdiv(n, d, (-1)^(n/d+1)*d)} \\ Andrew Howroyd, Sep 10 2019

G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^k * (1 + x^k) / (1 - x^k)^3.
a(n) = n * Sum_{d|n} (-1)^(n/d + 1) * d.
a(n) = n * A000593(n).
From Amiram Eldar, Dec 05 2022: (Start)
Multiplicative with a(2^e) = 2^e, and a(p^e) = p^e*(p^(e+1)-1)/(p-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/36 = 0.2741556... (A353908). (End)
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*(1-2^(2-s)). - Amiram Eldar, Jan 07 2023

A328368 Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of all positive integers <= n into k consecutive parts.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 02 2019

Column k lists k times every nonzero multiple of k in nondecreasing order.

Column k lists the partial sums of the k-th column of triangle A285914.

			Triangle begins:
   1;
   2;
   3,  2;
   4,  2;
   5,  4;
   6,  4,  3;
   7,  6,  3;
   8,  6,  3;
   9,  8,  6;
  10,  8,  6,  4;
  11, 10,  6,  4;
  12, 10,  9,  4;
  13, 12,  9,  4;
  14, 12,  9,  8;
  15, 14, 12,  8,  5;
  16, 14, 12,  8,  5;
  17, 16, 12,  8,  5;
  18, 16, 15, 12,  5;
  19, 18, 15, 12,  5;
  20, 18, 15, 12, 10;
  21, 20, 18, 12, 10,  6;
  22, 20, 18, 16, 10,  6;
  23, 22, 18, 16, 10,  6;
  24, 22, 21, 16, 10,  6;
  25, 24, 21, 16, 15,  6;
  26, 24, 21, 20, 15,  6;
  27, 26, 24, 20, 15, 12;
  28, 26, 24, 20, 15, 12, 7;
...

Row sums give A285899.
Row n has length A003056(n).
Column 1 gives A000027.
Column k starts with k in the row A000217(k).
Cf.  A052928, A196020, A204217, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A245579, A262612, A285900, A285914, A285891, A286000, A286001, A286013, A299765, A328361, A328365, A328371.

tt(n, k) = k*(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

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

A329321 a(n) is the total number of odd parts in all partitions of n into consecutive parts.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 2, 0, 3, 2, 2, 2, 2, 2, 6, 0, 2, 4, 2, 2, 6, 2, 2, 2, 5, 2, 6, 4, 2, 6, 2, 0, 6, 2, 8, 6, 2, 2, 6, 2, 2, 8, 2, 4, 14, 2, 2, 2, 5, 4, 6, 4, 2, 8, 10, 4, 6, 2, 2, 8, 2, 2, 14, 0, 10, 10, 2, 4, 6, 8, 2, 6, 2, 2, 14, 4, 10, 10, 2, 2, 11, 2, 2, 10, 10, 2, 6, 6, 2, 16

Offset: 1

Omar E. Pol, Nov 10 2019

nonn

a(n) = 0 if and only if n is an even power of 2.

			For n = 15 there are four partitions of 15 into consecutive part, they are [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1]. In total there are six odd parts, they are [15, 7, 5, 5, 3, 1], so a(15) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A204217 (total number of parts).

Cf. A000079, A001227, A002131, A237048, A245579, A285898, A285914, A286000, A286001, A299765, A328361, A328365, A329322.

A329321(n) = { my(i=2, t=(n%2)); n--; while(n>0, if(!(n%i), t += (((n/i)%2)+i)\2); n-=i; i++); t }; \\ (After David A. Corneth's program for A204217) - Antti Karttunen, Dec 09 2021

a(n) = A204217(n) - A329322(n).

A329322 a(n) is the total number of even parts in all partitions of n into consecutive parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 1, 3, 5, 1, 1, 4, 1, 4, 6, 3, 1, 2, 3, 3, 6, 4, 1, 7, 1, 1, 6, 3, 7, 6, 1, 3, 6, 4, 1, 7, 1, 5, 12, 3, 1, 2, 5, 6, 6, 5, 1, 9, 8, 4, 6, 3, 1, 9, 1, 3, 14, 1, 8, 9, 1, 5, 6, 9, 1, 7, 1, 3, 13, 5, 11, 10, 1, 4, 10, 3, 1, 9, 8, 3, 6, 6, 1, 18

Offset: 1

Omar E. Pol, Nov 10 2019

nonn

			For n = 15 there are four partitions of 15 into consecutive part, they are [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1]. In total there are five even parts, they are [8, 6, 4, 4, 2], so a(15) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A204217 (total number of parts).

Cf. A001227, A237048, A245579, A285898, A285914, A286000, A286001, A299765, A328361, A328365, A329321.

A329322(n) = { my(i=2, t=!(n%2)); n--; while(n>0, if(!(n%i), t += (!((n/i)%2)+i)\2); n-=i; i++); t }; \\ (After David A. Corneth's program for A204217) - Antti Karttunen, Dec 09 2021

a(n) = A204217(n) - A329321(n).

A309400 Irregular triangle read by rows in which row n lists in reverse order the partitions of n into equal parts.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 30 2019

The number of parts in row n equals sigma(n) = A000203(n), the sum of the divisors of n. More generally, the number of parts congruent to 0 (mod m) in row m*n equals sigma(n).
The number of parts greater than 1 in row n equals A001065(n), the sum of the 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 sum of row n equals n*A000005(n) = A038040(n).
Records in row n give the n-th row of A027750.
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 A056538(n,k) parts.
Column 1 gives A000012.
Right border gives A000027.

			Triangle begins:
[1];
[1,1], [2];
[1,1,1], [3];
[1,1,1,1], [2,2], [4];
[1,1,1,1,1], [5];
[1,1,1,1,1,1], [2,2,2], [3,3], [6];
[1,1,1,1,1,1,1], [7];
[1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [8];
[1,1,1,1,1,1,1,1,1], [3,3,3], [9];
[1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10];
[1,1,1,1,1,1,1,1,1,1,1], [11];
[1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3], [4,4,4], [6,6], [12];
[1,1,1,1,1,1,1,1,1,1,1,1,1], [13];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2], [7,7], [14];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [3,3,3,3,3], [5,5,5], [15];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2,2], [4,4,4,4], [8,8], [16];
...

Mirror of A244051.

Cf. A000005, A000012, A000027, A000203, A000217, A001065, A001227, A002131, A027750, A038040, A048050, A056538, A126988, A237593, A245579, A299765, A328365.

A334467 Square array read by antidiagonals upwards: T(n,k) is the sum of all parts of all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

Original entry on oeis.org

1, 4, 1, 6, 2, 1, 12, 6, 2, 1, 10, 4, 3, 2, 1, 24, 10, 8, 3, 2, 1, 14, 12, 5, 4, 3, 2, 1, 32, 14, 12, 10, 4, 3, 2, 1, 27, 8, 7, 6, 5, 4, 3, 2, 1, 40, 27, 16, 14, 12, 5, 4, 3, 2, 1, 22, 20, 18, 8, 7, 6, 5, 4, 3, 2, 1, 72, 22, 20, 18, 16, 14, 6, 5, 4, 3, 2, 1, 26, 24, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Omar E. Pol, May 05 2020

			Array begins:
     k  0   1   2   3   4   5   6   7   8   9  10
   n +------------------------------------------------
   1 |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   2 |  4,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
   3 |  6,  6,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...
   4 | 12,  4,  8,  4,  4,  4,  4,  4,  4,  4,  4, ...
   5 | 10, 10,  5, 10,  5,  5,  5,  5,  5,  5,  5, ...
   6 | 24, 12, 12,  6, 12,  6,  6,  6,  6,  6,  6, ...
   7 | 14, 14,  7, 14,  7, 14,  7,  7,  7,  7,  7, ...
   8 | 32,  8, 16,  8, 16,  8, 16,  8,  8,  8,  8, ...
   9 | 27, 27, 18, 18,  9, 18,  9, 18,  9,  9,  9, ...
  10 | 40, 20, 20, 10, 20, 20, 20, 10, 20, 10, 10, ...
...

Columns k: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), A334733 (k=5), A334953 (k=6).
Every diagonal starting with 1 gives A000027.
Sequences of number of parts related to column k: A000203 (k=0), A204217 (k=1), A066839 (k=2) (conjectured), A330889 (k=3), A334464 (k=4), A334732 (k=5), A334949 (k=6).
Sequences of number of partitions related to column k: A000005 (k=0), A001227 (k=1), A038548 (k=2), A117277 (k=3), A334461 (k=4), A334541 (k=5), A334948 (k=6).
Polygonal  numbers related to column k: A001477 (k=0), A000217 (k=1), A000290 (k=2), A000326 (k=3), A000384 (k=4), A000566 (k=5), A000567 (k=6).
Cf. A245579, A323345, A334466.

nmax = 13;
col[k_] := col[k] = CoefficientList[Sum[x^(n(k n - k + 2)/2 - 1)/(1 - x^n), {n, 1, nmax}] + O[x]^nmax, x];
T[n_, k_] := n col[k][[n]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 30 2020 *)

T(n,k) = n*A323345(n,k).

A367870 a(n) = Sum_{d|n, d odd} (n-d).

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 6, 7, 14, 14, 10, 20, 12, 20, 36, 15, 16, 41, 18, 34, 52, 32, 22, 44, 44, 38, 68, 48, 28, 96, 30, 31, 84, 50, 92, 95, 36, 56, 100, 74, 40, 136, 42, 76, 192, 68, 46, 92, 90, 119, 132, 90, 52, 176, 148, 104, 148, 86, 58, 216, 60, 92, 274, 63, 176, 216

Offset: 1

Wesley Ivan Hurt, Dec 03 2023

Total distance from n to each odd divisor of n.

			a(15) = 36. The total distance from 15 to each of its odd divisors is (15-1) + (15-3) + (15-5) + (15-15) = 36.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000593, A001227, A094471, A245579.

f:= proc(n) local x,d;
  x:= n/2^padic:-ordp(n,2);
  add(n-d, d = numtheory:-divisors(x))
end proc:
map(f, [$1..100]); # Robert Israel, Dec 04 2023

Table[DivisorSum[n, n-# &, OddQ], {n, 100}] (* Paolo Xausa, Mar 05 2024 *)

a(n) = sumdiv(n, d, if (d%2, n-d)); \\ Michel Marcus, Dec 04 2023

from math import prod
from sympy import factorint
def A367870(n):
    f = factorint(n>>(~n&n-1).bit_length())
    return n*prod(e+1 for e in f.values())-prod((p**(e+1)-1)//(p-1) for p,e in f.items()) # Chai Wah Wu, Dec 31 2023

a(n) = A245579(n) - A000593(n).

a(n) = n*A001227(n) - A000593(n).

A354009 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of equal parts, in nonincreasing order.

Original entry on oeis.org

1, 2, 3, 1, 1, 1, 4, 5, 1, 1, 1, 1, 1, 6, 2, 2, 2, 7, 1, 1, 1, 1, 1, 1, 1, 8, 9, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 2, 2, 2, 2, 2, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 4, 4, 4, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 2, 2, 2, 2, 2, 2, 2, 15, 5, 5, 5, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Jul 13 2022

The number of partitions in row n equals A001227(n), the number of odd divisors of n, and equals the number of partitions of n into consecutive parts, and equals the number of subparts in the symmetric representation of sigma(n).
The sum of row n equals A245579(n), the sum of all parts of all partitions of n into consecutive parts.
The length of row n equals A000593(n), the sum of the odd divisors of n.
Row n has length 1 if and only if n is a power of 2.
Is the right border the same as A006519?

			Triangle begins:
   [1];
   [2];
   [3], [1,1,1];
   [4];
   [5], [1,1,1,1,1];
   [6], [2,2,2];
   [7], [1,1,1,1,1,1,1];
   [8];
   [9], [3,3,3], [1,1,1,1,1,1,1,1,1];
  [10], [2,2,2,2,2];
  [11], [1,1,1,1,1,1,1,1,1,1,1];
  [12], [4,4,4];
  [13], [1,1,1,1,1,1,1,1,1,1,1,1,1];
  [14], [2,2,2,2,2,2,2];
  [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];
  ...
For n = 10, in row 10 there are two partitions, equaling the number of odd divisors of 10, they are [1, 5], and equaling the number of partitions of 10 into consecutive parts, they are [10], [4, 3, 2, 1], and equaling the number of subparts in the symmetric representation of sigma(10), they are [9, 9].
The sum of row 10 is [10] + [2 + 2 + 2 + 2 + 2] = 20 equaling the sum of all parts of all partitions of 10 into consecutive parts, that is [10] + [4 + 3 + 2 + 1] = 20.
The length of row 10 is equal to 6 equaling the sum of the odd divisors of 10, that is 1 + 5 = 6.

Subsequence of A244051.
The number of partitions in row n equals A001227(n).
Row lengths give A000593.
Row sums give A245579.
Column 1 gives A000027.
Cf. A000079, A000203, A006519, A237593, A279387 (subparts), A299765.

Table[ConstantArray[n/#, #] & /@ Select[Divisors[n], OddQ], {n, 15}] // Flatten (* Michael De Vlieger, Jul 15 2022 *)

row(n) = my(v=[]); fordiv(n, d, if ((n/d)%2, v = concat(v, vector(n/d, k, d)))); Vecrev(v); \\ Michel Marcus, Jul 16 2022

cp's OEIS Frontend

A326125 Expansion of Sum_{k>=1} k^2 * x^k / (1 + x^k)^2.

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A328368 Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of all positive integers <= n into k consecutive parts.

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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.

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A329321 a(n) is the total number of odd parts in all partitions of n into consecutive parts.

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A329322 a(n) is the total number of even parts in all partitions of n into consecutive parts.

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A309400 Irregular triangle read by rows in which row n lists in reverse order the partitions of n into equal parts.

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A334467 Square array read by antidiagonals upwards: T(n,k) is the sum of all parts of all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

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A367870 a(n) = Sum_{d|n, d odd} (n-d).

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