A245579 -id:A245579 - cp's OEIS frontend

A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

1, 2, 5, 4, 8, 10, 11, 8, 20, 16, 17, 20, 20, 22, 42, 16, 26, 40, 29, 32, 58, 34, 35, 40, 53, 40, 74, 44, 44, 84, 47, 32, 90, 52, 94, 80, 56, 58, 106, 64, 62, 116, 65, 68, 174, 70, 71, 80, 102, 106, 138, 80, 80, 148, 146, 88, 154, 88, 89, 168, 92, 94, 241, 64, 172

Offset: 1

Ilya Gutkovskiy, Oct 07 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A328203, Sequence Machine.

Cf. A000005, A000079 (fixed points), A000203, A001227, A002131, A003602, A006519, A006918, A026741, A038040, A109168, A113415, A140472, A152211, A193356, A245579, A349353 (Dirichlet inverse), A349354.

Cf. also A347957.

a:=[]; for k in [1..65] do if IsOdd(k) then a[k]:=(k * #Divisors(k) + DivisorSigma(1,k)) / 2; else a[k]:=(k * (#Divisors(k) - #Divisors(k div 2)) + DivisorSigma(1,k) - DivisorSigma(1,k div 2)) / 2;  end if; end for; a; // Marius A. Burtea, Oct 07 2019

nmax = 65; CoefficientList[Series[Sum[k x^k/(1 - x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, (n Mod[#, 2] + Boole[OddQ[n/#]] #)/2 &]; Table[a[n], {n, 1, 65}]

A328203(n) = if(n%2,(1/2)*(sigma(n)+(n*numdiv(n))),2*A328203(n/2)); \\ Antti Karttunen, Nov 13 2021

a(n) = (n * d(n) + sigma(n)) / 2 if n odd, (n * (d(n) - d(n/2)) + sigma(n) - sigma(n/2)) / 2 if n even.
a(n) = (n * A001227(n) + A002131(n)) / 2.
a(2*n) = 2 * a(n).
From Antti Karttunen, Nov 13 2021: (Start)
The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. Both are easy to prove:
a(n) = Sum_{d|n} A003602(d) * A026741(n/d).
a(n) = Sum_{d|n} A109168(d) * A193356(n/d), where A109168(d) = A140472(d) = (d+A006519(d))/2.
(End)

A328362 Triangle read by rows: T(n,k) is the sum of all parts k in all partitions of n into consecutive parts, (1 <= k <= n).

Original entry on oeis.org

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

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 n.

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

			Triangle begins:
1;
0, 2;
1, 2, 3;
0, 0, 0, 4;
0, 2, 3, 0, 5;
1, 2, 3, 0, 0, 6;
0, 0, 3, 4, 0, 0, 7;
0, 0, 0, 0, 0, 0, 0, 8;
0, 2, 3, 8, 5, 0, 0, 0, 9;
1, 2, 3, 4, 0, 0, 0, 0, 0, 10;
0, 0, 0, 0, 5, 6, 0, 0, 0,  0, 11;
0, 0, 3, 4, 5, 0, 0, 0, 0,  0,  0, 12;
0, 0, 0, 0, 0, 6, 7, 0, 0,  0,  0,  0, 13;
0, 2, 3, 4, 5, 0, 0, 0, 0,  0,  0,  0,  0, 14;
1, 2, 3, 8,10, 6, 7, 8, 0,  0,  0,  0,  0,  0, 15;
0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0, 16;
...
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, 2, 3, 8, 5, 0, 0, 0, 9].

Row sums give A245579.
Column 1 gives A010054, n => 1.
Leading diagonal gives A000027.
Cf. A000079, A001227, A065091, A138785, A204217, A237048, A237593, A266531, A285898, A285899, A285900, A285914, A286000, A286001, A299765, A328361.

T(n,k) = k*A328361(n,k).

A334953 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 6.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 42, 44, 23, 72, 25, 52, 54, 56, 29, 90, 31, 64, 66, 68, 35, 108, 37, 76, 78, 120, 41, 126, 43, 132, 90, 92, 47, 192, 49, 100, 102, 156, 53, 162, 55, 168, 114, 116, 59, 240, 61, 124, 126, 192, 130, 198, 67, 204, 138, 210

Offset: 1

Omar E. Pol, May 27 2020

nonn

The one-part partition n = n is included in the count.

			For n = 24 there are three partitions of 24 into consecutive parts that differ by 6, including 24 as a valid partition. They are [24], [15, 9] and [14, 8, 2]. The sum of all parts is [24] + [15 + 9] + [14 + 8 + 2] = 72, so a(24) = 72.

Sequences of the same family where the parts differs by k are: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), A334733 (k=5), this sequence (k=6).

Cf. A334946, A334947, A334948, A334949.

a(n) = n*A334948(n).

A286014 Sum of smallest parts of all partitions of n into consecutive parts.

Original entry on oeis.org

1, 2, 4, 4, 7, 7, 10, 8, 15, 11, 16, 15, 19, 16, 27, 16, 25, 26, 28, 22, 38, 26, 34, 31, 40, 31, 50, 29, 43, 49, 46, 32, 62, 41, 59, 48, 55, 46, 74, 46, 61, 67, 64, 46, 94, 56, 70, 63, 77, 69, 98, 55, 79, 85, 92, 61, 110, 71, 88, 93, 91, 76, 131, 64, 110, 103

Offset: 1

Omar E. Pol, Apr 30 2017

nonn

If n is a power of 2 then a(n) = n, the same as A286015(n).

Conjecture: this is also the row sums of A211343.

			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 sum of the smallest parts is 15 + 7 + 4 + 1 = 27, so a(15) = 27.

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

Cf. A000079, A046746, A204217, A211343, A245579, A286015.

Table[Total[Select[IntegerPartitions@ n, Or[Length@ # == 1, Union@ Differences@ # == {-1}] &][[All, -1]]], {n, 66}] (* Michael De Vlieger, Jul 21 2017 *)

More terms from Alois P. Heinz, May 01 2017

A286015 Sum of largest parts of all partitions of n into consecutive parts.

Original entry on oeis.org

1, 2, 5, 4, 8, 9, 11, 8, 18, 14, 17, 17, 20, 19, 34, 16, 26, 31, 29, 26, 46, 29, 35, 33, 45, 34, 58, 35, 44, 58, 47, 32, 70, 44, 70, 57, 56, 49, 82, 50, 62, 78, 65, 53, 114, 59, 71, 65, 84, 76, 106, 62, 80, 98, 106, 67, 118, 74, 89, 106, 92, 79, 153, 64, 124

Offset: 1

Omar E. Pol, Apr 30 2017

nonn

If n is a power of 2 then a(n) = n, the same as A286014(n).

Conjecture: this is also the row sums of A286013.

			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 sum of the largest parts is 15 + 8 + 6 + 5 = 34, so a(15) = 34.

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

Cf. A000079, A006128, A046746, A204217, A211343, A245579, A286013, A286014.

Table[Total[Select[IntegerPartitions@ n, Or[Length@ # == 1, Union@ Differences@ # == {-1}] &][[All, 1]]], {n, 65}] (* Michael De Vlieger, Jul 21 2017 *)

More terms from Alois P. Heinz, May 01 2017

A334463 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 3.

Original entry on oeis.org

1, 2, 3, 4, 10, 6, 14, 8, 18, 10, 22, 24, 26, 14, 45, 16, 34, 36, 38, 20, 63, 44, 46, 48, 50, 52, 81, 28, 58, 90, 62, 32, 99, 68, 105, 72, 74, 76, 117, 80, 82, 126, 86, 44, 180, 92, 94, 96, 98, 150, 204, 52, 106, 162, 165, 56, 228, 116, 118, 180, 122, 124, 252, 64, 195, 198, 134, 68, 276, 280, 142, 144

Offset: 1

Omar E. Pol, May 05 2020

nonn

The one-part partition n = n is included in the count.

			For n = 21 there are three partitions of 21 into consecutive parts that differ by 3, including 21 as a valid partition. They are [21], [12, 9] and [10, 7, 4]. The sum of the parts is [21] + [12 + 9] + [10 + 7 + 4] = 63, the same as 3*21 = 63, since there are three of these partitions of 21, so a(21) = 63.

Cf. A038040, A245579, A330887, A330888, A330889.

Sequences of the same family where the parts differs by k are: A038040 (k=0), A245579 (k=1), A060872 (k=2), this sequence (k=3), A327262 (k=4).

a(n) = n*A117277(n).

A334733 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 14, 8, 18, 10, 22, 12, 26, 14, 30, 16, 34, 36, 38, 20, 63, 22, 46, 48, 50, 26, 81, 28, 58, 60, 62, 32, 99, 68, 70, 72, 74, 76, 117, 40, 82, 126, 86, 44, 135, 92, 94, 96, 98, 100, 153, 52, 106, 162, 165, 56, 171, 116, 118, 180, 122, 124, 189, 64, 195, 198, 134, 68, 207, 210

Offset: 1

Omar E. Pol, May 09 2020

nonn

The one-part partition n = n is included in the count.

			For n = 27 there are three partitions of 27 into consecutive parts that differ by 5, including 27 as a valid partition. They are [27], [16, 11] and [14, 9, 4]. The sum of all parts is [27] + [16 + 11] + [14 + 9 + 4] = 81, so a(27) = 81.

Sequences of the same family where the parts differs by k are: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), this sequence (k=5).

Cf. A334465, A334541.

a(n) = n*A334541(n).

A352257 Sum of all parts of all partitions of n into an odd number of consecutive parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 7, 8, 18, 10, 11, 24, 13, 14, 45, 16, 17, 36, 19, 40, 42, 22, 23, 48, 50, 26, 54, 56, 29, 90, 31, 32, 66, 34, 105, 72, 37, 38, 78, 80, 41, 126, 43, 44, 180, 46, 47, 96, 98, 100, 102, 52, 53, 162, 110, 112, 114, 58, 59, 180, 61, 62, 252, 64, 130, 198

Offset: 1

Omar E. Pol, Mar 09 2022

nonn

a(n) is n times the number of partitions of n into an odd number of consecutive parts.

			For n = 15 the partitions of 15 into an odd number of consecutive parts are [15], [6, 5, 4] and [5, 4, 3, 2, 1], so a(15) = 15 + 6 + 5 + 4 + 5 + 4 + 3 + 2 + 1 = 15*3 = 45.

Row sums of A352499.

Cf. A066186, A082647, A237048, A245579, A286000, A286001, A299765, A351824, A352505.

a(n) = my(q = sqrt(2*n)); n*sumdiv(n, d, (d%2) && (d < q)); \\ Michel Marcus, Mar 11 2022; after A082647

a(n) = n*A082647(n).

a(n) = A245579(n) - A352505(n). - Omar E. Pol, Mar 19 2022

A352499 Irregular triangle read by rows: T(n,k) is the sum of all parts of the partition of n into consecutive parts that contains 2*k-1 parts, and the first element of the column k is in row A000384(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 0, 8, 0, 9, 9, 10, 0, 11, 0, 12, 12, 13, 0, 14, 0, 15, 15, 15, 16, 0, 0, 17, 0, 0, 18, 18, 0, 19, 0, 0, 20, 0, 20, 21, 21, 0, 22, 0, 0, 23, 0, 0, 24, 24, 0, 25, 0, 25, 26, 0, 0, 27, 27, 0, 28, 0, 0, 28, 29, 0, 0, 0, 30, 30, 30, 0, 31, 0, 0, 0, 32, 0, 0, 0

Offset: 1

Omar E. Pol, Mar 19 2022

This triangle is formed from the odd-indexed columns of the triangle A285891.

			Triangle begins:
   1;
   2;
   3;
   4;
   5;
   6,  6;
   7,  0;
   8,  0;
   9,  9;
  10,  0;
  11,  0;
  12, 12;
  13,  0;
  14,  0;
  15, 15, 15;
  16,  0,  0;
  17,  0,  0;
  18, 18,  0;
  19,  0,  0;
  20,  0, 20;
  21, 21,  0;
  22,  0,  0;
  23,  0,  0;
  24, 24,  0;
  25,  0, 25;
  26,  0,  0;
  27, 27,  0;
  28,  0,  0, 28;
  ...
For n = 21 the partitions of 21 into on odd number of consecutive parts are [21] and [8, 7, 6], so T(21,1) = 1 and T(21,2) = 8 + 7 + 6 = 21. There is no partition of 21 into five consecutive parts so T(21,3) = 0.

Paolo Xausa, Table of n, a(n) for n = 1..10490 (rows 1..800 of triangle, flattened).

Row sums give A352257.
Row n has A351846(n) terms.
The number of nonzero terms in row n equals A082647(n).
Cf. A000384, A237048, A245579, A285891, A299765, A351824, A352425.

A352499[rowmax_]:=Table[Boole[Divisible[n,2k-1]]n,{n,rowmax},{k,Floor[(Sqrt[8n+1]+1)/4]}];A352499[50] (* Paolo Xausa, Apr 12 2023 *)

T(n,k) = n*A351824(n,k).

T(n,k) = n*[(2*k-1)|n], where 1 <= k <= floor((sqrt(8*n+1)+1)/4) and [] is the Iverson bracket. - Paolo Xausa, Apr 12 2023

A352505 Sum of all parts of all partitions of n into an even number of consecutive parts.

Original entry on oeis.org

0, 0, 3, 0, 5, 0, 7, 0, 9, 10, 11, 0, 13, 14, 15, 0, 17, 18, 19, 0, 42, 22, 23, 0, 25, 26, 54, 0, 29, 30, 31, 0, 66, 34, 35, 36, 37, 38, 78, 0, 41, 42, 43, 44, 90, 46, 47, 0, 49, 50, 102, 52, 53, 54, 110, 0, 114, 58, 59, 60, 61, 62, 126, 0, 130, 66, 67, 68, 138, 70, 71, 0

Offset: 1

Omar E. Pol, Mar 19 2022

nonn

			For n = 21 the partitions of 21 into an even number of consecutive parts are [11, 10] and [6, 5, 4, 3, 2, 1], so a(21) = 11 + 10 + 6 + 5 + 4 + 3 + 2 + 1 = 21*2 = 42.

Indices of zero terms give A082662.
Indices of nonzero terms give A281005.
Cf. A066186, A131576, A245579, A285891, A286000, A286001, A299765, A352257.

a(n) = n*A131576(n).

a(n) = A245579(n) - A352257(n).

cp's OEIS Frontend

A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A328362 Triangle read by rows: T(n,k) is the sum of all parts k in all partitions of n into consecutive parts, (1 <= k <= n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A334953 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 6.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A286014 Sum of smallest parts of all partitions of n into consecutive parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A286015 Sum of largest parts of all partitions of n into consecutive parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A334463 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 3.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A334733 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 5.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A352257 Sum of all parts of all partitions of n into an odd number of consecutive parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A352499 Irregular triangle read by rows: T(n,k) is the sum of all parts of the partition of n into consecutive parts that contains 2*k-1 parts, and the first element of the column k is in row A000384(k).

Views

Author

Keywords