A360674 -id:A360674 - cp's OEIS frontend

A360672 Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 3, 1, 0, 1, 0, 2, 3, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 0, 0, 3, 6, 4, 1, 0, 1, 0, 0, 1, 7, 7, 5, 1, 0, 1, 0, 0, 1, 4, 8, 10, 5, 1, 0, 1, 0, 0, 0, 3, 6, 14, 11, 6, 1, 0, 1, 0, 0, 0, 1, 5, 12, 16, 14, 6, 1, 0

Offset: 0

Gus Wiseman, Feb 27 2023

Also the number of integer partitions of n whose right half (inclusive) sums to n-k.

			Triangle begins:
  1
  1  0
  1  1  0
  1  1  1  0
  1  0  3  1  0
  1  0  2  3  1  0
  1  0  1  4  4  1  0
  1  0  0  3  6  4  1  0
  1  0  0  1  7  7  5  1  0
  1  0  0  1  4  8 10  5  1  0
  1  0  0  0  3  6 14 11  6  1  0
  1  0  0  0  1  5 12 16 14  6  1  0
  1  0  0  0  1  2 12 14 23 16  7  1  0
  1  0  0  0  0  2  7 13 24 27 19  7  1  0
  1  0  0  0  0  1  5  9 24 30 35 21  8  1  0
  1  0  0  0  0  1  3  7 17 31 42 40 25  8  1  0
  1  0  0  0  0  0  2  4 16 23 46 51 51 27  9  1  0
  1  0  0  0  0  0  1  3 10 21 37 57 69 57 31  9  1  0
  1  0  0  0  0  0  1  2  7 15 34 47 83 81 69 34 10  1  0
For example, row n = 9 counts the following partitions:
  (9)  .  .  (333)  (432)        (54)        (63)      (72)    (81)
                    (441)        (522)       (621)     (711)
                    (22221)      (531)       (3321)    (4311)
                    (111111111)  (3222)      (4221)    (5211)
                                 (32211)     (33111)   (6111)
                                 (2211111)   (42111)
                                 (3111111)   (51111)
                                 (21111111)  (222111)
                                             (321111)
                                             (411111)
For example, the partition y = (3,2,2,1,1) has left half (exclusive) (3,2), with sum 5, so y is counted under T(9,5).

Row sums are A000041.
Column sums are A360673, inclusive A360671.
The central diagonal T(2n,n) is A360674, ranks A360953.
The left inclusive version is A360675 with rows reversed.
A008284 counts partitions by length.
A359893 and A359901 count partitions by median.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A027193, A237363, A307683, A325347, A360005, A360071, A360254, A360616, A360682, A360686.

Table[Length[Select[IntegerPartitions[n], Total[Take[#,Floor[Length[#]/2]]]==k&]],{n,0,10},{k,0,n}]

A360675 Triangle read by rows where T(n,k) is the number of integer partitions of n whose right half (exclusive) sums to k, where k ranges from 0 to n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0, 1, 3, 3, 0, 0, 0, 1, 3, 5, 2, 0, 0, 0, 1, 4, 6, 4, 0, 0, 0, 0, 1, 4, 9, 5, 3, 0, 0, 0, 0, 1, 5, 10, 10, 4, 0, 0, 0, 0, 0, 1, 5, 13, 12, 9, 2, 0, 0, 0, 0, 0, 1, 6, 15, 18, 11, 5, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Feb 27 2023

Also the number of integer partitions of n whose left half (inclusive) sums to n-k.

			Triangle begins:
  1
  1  0
  1  1  0
  1  2  0  0
  1  2  2  0  0
  1  3  3  0  0  0
  1  3  5  2  0  0  0
  1  4  6  4  0  0  0  0
  1  4  9  5  3  0  0  0  0
  1  5 10 10  4  0  0  0  0  0
  1  5 13 12  9  2  0  0  0  0  0
  1  6 15 18 11  5  0  0  0  0  0  0
  1  6 18 22 20  6  4  0  0  0  0  0  0
  1  7 20 29 26 13  5  0  0  0  0  0  0  0
  1  7 24 34 37 19 11  2  0  0  0  0  0  0  0
  1  8 26 44 46 30 16  5  0  0  0  0  0  0  0  0
  1  8 30 50 63 40 27  8  4  0  0  0  0  0  0  0  0
  1  9 33 61 75 61 36 15  6  0  0  0  0  0  0  0  0  0
  1  9 37 70 96 75 61 21 12  3  0  0  0  0  0  0  0  0  0
For example, row n = 9 counts the following partitions:
  (9)  (81)   (72)     (63)       (54)
       (441)  (432)    (333)      (3222)
       (531)  (522)    (3321)     (21111111)
       (621)  (4311)   (4221)     (111111111)
       (711)  (5211)   (22221)
              (6111)   (222111)
              (32211)  (321111)
              (33111)  (411111)
              (42111)  (2211111)
              (51111)  (3111111)
For example, the partition y = (3,2,2,1,1) has right half (exclusive) (1,1), with sum 2, so y is counted under T(9,2).

The central diagonal T(2n,n) is A000005.
Row sums are A000041.
Diagonal sums are A360671, exclusive A360673.
The right inclusive version is A360672 with rows reversed.
The left version has central diagonal A360674, ranks A360953.
A008284 counts partitions by length.
A359893 and A359901 count partitions by median.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A027193, A237363, A307683, A325347, A360005, A360071, A360254, A360616.

Table[Length[Select[IntegerPartitions[n], Total[Take[#,-Floor[Length[#]/2]]]==k&]],{n,0,18},{k,0,n}]

A360673 Number of multisets of positive integers whose right half (exclusive) sums to n.

Original entry on oeis.org

1, 2, 7, 13, 27, 37, 73, 89, 156, 205, 315, 387, 644, 749, 1104, 1442, 2015, 2453, 3529, 4239, 5926, 7360, 9624, 11842, 16115, 19445, 25084, 31137, 39911, 48374, 62559, 75135, 95263, 115763, 143749, 174874, 218614, 261419, 321991, 388712, 477439, 569968, 698493

Offset: 0

Gus Wiseman, Mar 04 2023

nonn

			The a(0) = 1 through a(3) = 13 multisets:
  {}  {1,1}    {1,2}        {1,3}
      {1,1,1}  {2,2}        {2,3}
               {1,1,2}      {3,3}
               {1,2,2}      {1,1,3}
               {2,2,2}      {1,2,3}
               {1,1,1,1}    {1,3,3}
               {1,1,1,1,1}  {2,2,3}
                            {2,3,3}
                            {3,3,3}
                            {1,1,1,2}
                            {1,1,1,1,2}
                            {1,1,1,1,1,1}
                            {1,1,1,1,1,1,1}
For example, the multiset y = {1,1,1,1,2} has right half (exclusive) {1,2}, with sum 3, so y is counted under a(3).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The inclusive version is A360671.
Column sums of A360672.
The case of sets is A360954, inclusive A360955.
The even-length case is A360956.
A359893 and A359901 count partitions by median.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000041, A360616, A360617, A360674, A360675, A360953.

Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], Total[Take[#,Floor[Length[#]/2]]]==k&]],{k,0,15}]

seq(n)={my(s=1 + O(x*x^n), p=s); for(k=1, n, s += p*x^k*(2-x^k)/(1-x^k + O(x*x^(n-k)))^(k+2); p /= 1 - x^k); Vec(s)} \\ Andrew Howroyd, Mar 11 2023

G.f.: 1 + Sum_{k>=1} x^k*(2 - x^k)/((1 - x^k)^(k+2) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023

Terms a(21) and beyond from Andrew Howroyd, Mar 11 2023

A360676 Sum of the left half (exclusive) of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 04 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 810 are {1,2,2,2,2,3}, with left half (exclusive) {1,2,2}, so a(810) = 5.
The prime indices of 3675 are {2,3,3,4,4}, with left half (exclusive) {2,3}, so a(3675) = 5.

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

Positions of 0's are 1 and A000040.
Positions of first appearances are 1 and A001248.
These partitions are counted by A360675, right A360672.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360616 gives half of bigomega (exclusive), inclusive A360617.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A026424, A280076, A359912, A360006, A360457, A360674.

f:= proc(n) local F,i,t;
  F:= [seq(numtheory:-pi(t[1])$t[2], t = sort(ifactors(n)[2],(a,b) -> a[1] < b[1]))];
  add(F[i],i=1..floor(nops(F)/2))
end proc:
map(f, [$1..100]); # Robert Israel, Feb 02 2025

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Take[prix[n],Floor[Length[prix[n]]/2]]],{n,100}]

A360676(n) + A360679(n) = A001222(n).

A360677(n) + A360678(n) = A001222(n).

A360616 Half the number of prime factors of n (counted with multiplicity, A001222), rounded down.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 2, 0, 1, 1, 3, 1, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 2, 2, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 08 2023

nonn

			The prime indices of 378 are {1,2,2,2,4}, so a(378) = floor(5/2) = 2.

Positions of 0's are 1 and A000040.
Positions of first appearances are A000302 = 2^(2k) for k >= 0.
Positions of 1's are A168645.
Rounding up instead of down gives A360617.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A001248, A026424, A280076, A359912, A360006, A360457, A360674.

Mathematica
```
Table[Floor[PrimeOmega[n]/2],{n,100}]
```

A360953 Numbers whose right half of prime indices (exclusive) adds up to half the total sum of prime indices.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 30, 48, 49, 63, 64, 70, 81, 108, 121, 154, 165, 169, 192, 256, 270, 273, 286, 289, 325, 361, 442, 529, 561, 567, 595, 625, 646, 675, 729, 741, 750, 768, 841, 874, 931, 961, 972, 1024, 1045, 1173, 1334, 1369, 1495, 1575, 1653, 1681, 1750

Offset: 1

Gus Wiseman, Mar 09 2023

nonn

Also numbers whose left half of prime indices (inclusive) adds up to half the total sum of prime indices.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    12: {1,1,2}
    16: {1,1,1,1}
    25: {3,3}
    30: {1,2,3}
    48: {1,1,1,1,2}
    49: {4,4}
    63: {2,2,4}
    64: {1,1,1,1,1,1}
    70: {1,3,4}
    81: {2,2,2,2}
   108: {1,1,2,2,2}
For example, the prime indices of 1575 are {2,2,3,3,4}, with right half (exclusive) {3,4}, with sum 7, and the total sum of prime indices is 14, so 1575 is in the sequence.

The left version is A056798.
The inclusive version is A056798.
These partitions are counted by A360674.
The left inclusive version is A360953 (this sequence).
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000005, A000040, A001248, A026424, A359912, A360006, A360616, A360617, A360671, A360673.

Select[Range[100],With[{w=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[Take[w,-Floor[Length[w]/2]]]==Total[w]/2]&]

A360955 Number of finite sets of positive integers whose right half (inclusive) sums to n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 11, 12, 19, 20, 31, 33, 49, 51, 77, 79, 112, 124, 165, 177, 247, 260, 340, 388, 480, 533, 693, 747, 925, 1078, 1271, 1429, 1772, 1966, 2331, 2705, 3123, 3573, 4245, 4737, 5504, 6424, 7254, 8256, 9634, 10889, 12372, 14251, 16031, 18379

Offset: 0

Gus Wiseman, Mar 09 2023

nonn

			The a(1) = 1 through a(8) = 12 sets:
  {1}  {2}    {3}    {4}    {5}      {6}      {7}        {8}
       {1,2}  {1,3}  {1,4}  {1,5}    {1,6}    {1,7}      {1,8}
              {2,3}  {2,4}  {2,5}    {2,6}    {2,7}      {2,8}
                     {3,4}  {3,5}    {3,6}    {3,7}      {3,8}
                            {4,5}    {4,6}    {4,7}      {4,8}
                            {1,2,3}  {5,6}    {5,7}      {5,8}
                                     {1,2,4}  {6,7}      {6,8}
                                              {1,2,5}    {7,8}
                                              {1,3,4}    {1,2,6}
                                              {2,3,4}    {1,3,5}
                                              {1,2,3,4}  {2,3,5}
                                                         {1,2,3,5}
For example, the set y = {2,3,5} has right half (inclusive) {3,5}, with sum 8, so y is counted under a(8).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for multisets is A360671, exclusive A360673.
The exclusive version is A360954.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000009, A072233, A359893, A359901, A360674, A360956.

Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], UnsameQ@@#&&Total[Take[#,Ceiling[Length[#]/2]]]==k&]],{k,0,15}]

\\ P(n,k) is A072233(n,k).
P(n,k)=polcoef(1/prod(k=1, k, 1 - x^k + O(x*x^n)), n)
a(n)=if(n==0, 1, sum(w=1, sqrt(n), my(t=binomial(w,2)); sum(h=w, (n-t)\w, binomial(h, w) * P(n-w*h-t, w-1)))) \\ Andrew Howroyd, Mar 13 2023

a(n) = Sum_{w>=1} Sum_{h=w..floor((n-binomial(w,2))/w)} binomial(h,w) * A072233(n - w*h - binomial(w,2), w-1) for n > 0. - Andrew Howroyd, Mar 13 2023

Terms a(16) and beyond from Andrew Howroyd, Mar 13 2023

A360956 Number of finite even-length multisets of positive integers whose right half sums to n.

Original entry on oeis.org

1, 1, 3, 5, 10, 13, 26, 31, 55, 73, 112, 140, 233, 276, 405, 539, 750, 931, 1327, 1627, 2259, 2839, 3708, 4624, 6237, 7636, 9823, 12275, 15715, 19227, 24735, 30000, 37930, 46339, 57574, 70374, 87704, 105606, 129998, 157417, 193240, 231769, 283585, 339052, 411682, 493260

Offset: 0

Gus Wiseman, Mar 09 2023

nonn

			The a(1) = 1 through a(5) = 13 multisets:
  {1,1}  {1,2}      {1,3}          {1,4}              {1,5}
         {2,2}      {2,3}          {2,4}              {2,5}
         {1,1,1,1}  {3,3}          {3,4}              {3,5}
                    {1,1,1,2}      {4,4}              {4,5}
                    {1,1,1,1,1,1}  {1,1,1,3}          {5,5}
                                   {1,1,2,2}          {1,1,1,4}
                                   {1,2,2,2}          {1,1,2,3}
                                   {2,2,2,2}          {1,2,2,3}
                                   {1,1,1,1,1,2}      {2,2,2,3}
                                   {1,1,1,1,1,1,1,1}  {1,1,1,1,1,3}
                                                      {1,1,1,1,2,2}
                                                      {1,1,1,1,1,1,1,2}
                                                      {1,1,1,1,1,1,1,1,1,1}
For example, the multiset y = {1,2,2,3} has right half {2,3}, with sum 5, so y is counted under a(5).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

This is the even-length case of A360671 and A360673.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000041, A360674, A360954, A360955.

Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], EvenQ[Length[#]]&&Total[Take[#,Length[#]/2]]==k&]],{k,0,15}]

seq(n)={my(s=1 + O(x*x^n), p=s); for(k=1, n, s += p*x^k/(1-x^k + O(x*x^(n-k)))^(k+1); p /= 1 - x^k); Vec(s)} \\ Andrew Howroyd, Mar 11 2023

G.f.: 1 + Sum_{k>=1} x^k/((1 - x^k)^(k+1) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023

Terms a(16) and beyond from Andrew Howroyd, Mar 11 2023

cp's OEIS Frontend

A360672 Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.

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A360675 Triangle read by rows where T(n,k) is the number of integer partitions of n whose right half (exclusive) sums to k, where k ranges from 0 to n.

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A360673 Number of multisets of positive integers whose right half (exclusive) sums to n.

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A360676 Sum of the left half (exclusive) of the prime indices of n.

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Maple

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A360616 Half the number of prime factors of n (counted with multiplicity, A001222), rounded down.

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A360953 Numbers whose right half of prime indices (exclusive) adds up to half the total sum of prime indices.

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Mathematica

A360955 Number of finite sets of positive integers whose right half (inclusive) sums to n.

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Examples

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Programs

Mathematica

PARI

Formula

Extensions

A360956 Number of finite even-length multisets of positive integers whose right half sums to n.

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