A361201 -id:A361201 - cp's OEIS frontend

A360617 Half the number of prime factors of n (counted with multiplicity, A001222), rounded up.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 2, 1, 1, 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) = ceiling(5/2) = 3.

Positions of 0's and 1's are 1 and A037143.
Positions of first appearances are A081294.
Rounding down instead of up gives A360616.
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. A000040, A000302, A001248, A026424, A168645, A359912, A360006, A360457.

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

A360674 Number of integer partitions of 2n whose left half (exclusive) and right half (inclusive) both sum to n.

Original entry on oeis.org

1, 1, 3, 4, 7, 6, 12, 9, 16, 15, 21, 16, 34, 22, 33, 36, 47, 36, 62, 44, 75, 68, 78, 68, 120, 93, 113, 117, 151, 122, 195, 148, 209, 197, 220, 226, 315, 249, 304, 309, 402, 332, 463, 387, 496, 515, 539, 514, 712, 609, 738, 723, 845, 774, 983, 914, 1111

Offset: 0

Gus Wiseman, Mar 04 2023

nonn

Of course, only one of the two conditions is necessary.

			The a(1) = 1 through a(6) = 12 partitions:
  (11)  (22)    (33)      (44)        (55)          (66)
        (211)   (321)     (422)       (532)         (633)
        (1111)  (21111)   (431)       (541)         (642)
                (111111)  (2222)      (32221)       (651)
                          (22211)     (211111111)   (3333)
                          (2111111)   (1111111111)  (33222)
                          (11111111)                (33321)
                                                    (42222)
                                                    (222222)
                                                    (2222211)
                                                    (21111111111)
                                                    (111111111111)
For example, the partition y = (3,2,2,2,1) has halves (3,2) and (2,2,1), both with sum 5, so y is counted under a(5).

The even-length case is A000005.
Central diagonal of A360672.
These partitions have ranks A360953.
A008284 counts partitions by length, row sums A000041.
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. A237363, A360254, A360671, A360673, A360682.

Table[Length[Select[IntegerPartitions[2n], Total[Take[#,Floor[Length[#]/2]]]==n&]],{n,0,15}]

def accel_asc(n):
    a = [0 for i in range(n + 1)]
    k = 1
    y = n - 1
    while k != 0:
        x = a[k - 1] + 1
        k -= 1
        while 2 * x <= y:
            a[k] = x
            y -= x
            k += 1
        l = k + 1
        while x <= y:
            a[k] = x
            a[l] = y
            yield a[:k + 2]
            x += 1
            y -= 1
        a[k] = x + y
        y = x + y - 1
        yield a[:k + 1]
for y in range(1000):
    num = 0
    for x in accel_asc(2*y):
        stop = len(x)//2+1
        if len(x) % 2 == 0:
            stop -= 1
        right = x[0:stop]
        left = x[stop:]
        if sum(right) == sum(left):
            num += 1
    print(y,num)
# David Consiglio, Jr., Mar 09 2023

a(n) = A360672(2n,n).

More terms from David Consiglio, Jr., Mar 09 2023

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]&]

A360954 Number of finite sets of positive integers whose right half (exclusive) sums to n.

Original entry on oeis.org

1, 0, 1, 3, 6, 10, 15, 22, 29, 41, 50, 70, 81, 113, 126, 176, 191, 264, 286, 389, 413, 569, 595, 798, 861, 1121, 1187, 1585, 1653, 2132, 2334, 2906, 3111, 4006, 4234, 5252, 5818, 6995, 7620, 9453, 10102, 12165, 13663, 15940, 17498, 21127, 22961, 26881, 30222, 34678, 38569

Offset: 0

Gus Wiseman, Mar 09 2023

nonn

			The a(2) = 1 through a(7) = 22 sets:
  {1,2}  {1,3}    {1,4}    {1,5}    {1,6}    {1,7}
         {2,3}    {2,4}    {2,5}    {2,6}    {2,7}
         {1,2,3}  {3,4}    {3,5}    {3,6}    {3,7}
                  {1,2,4}  {4,5}    {4,6}    {4,7}
                  {1,3,4}  {1,2,5}  {5,6}    {5,7}
                  {2,3,4}  {1,3,5}  {1,2,6}  {6,7}
                           {1,4,5}  {1,3,6}  {1,2,7}
                           {2,3,5}  {1,4,6}  {1,3,7}
                           {2,4,5}  {1,5,6}  {1,4,7}
                           {3,4,5}  {2,3,6}  {1,5,7}
                                    {2,4,6}  {1,6,7}
                                    {2,5,6}  {2,3,7}
                                    {3,4,6}  {2,4,7}
                                    {3,5,6}  {2,5,7}
                                    {4,5,6}  {2,6,7}
                                             {3,4,7}
                                             {3,5,7}
                                             {3,6,7}
                                             {4,5,7}
                                             {4,6,7}
                                             {5,6,7}
                                             {1,2,3,4}
For example, the set y = {1,2,3,4} has right half (exclusive) {3,4}, with sum 7, so y is counted under a(7).

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

The version for multisets is A360673, inclusive A360671.
The inclusive version is A360955.
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, A360956.

Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k],UnsameQ@@#&&Total[Take[#,Floor[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+1, (n-t)\w, binomial(h, w+1) * P(n-w*h-t, w-1)))) \\ Andrew Howroyd, Mar 13 2023

a(n) = Sum_{w>=1} Sum_{h=w+1..floor((n-binomial(w,2))/w)} binomial(h,w+1) * 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

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

A360617 Half the number of prime factors of n (counted with multiplicity, A001222), rounded up.

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A360674 Number of integer partitions of 2n whose left half (exclusive) and right half (inclusive) both sum to n.

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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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Author

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Comments

Examples

Crossrefs

Programs

Mathematica

A360954 Number of finite sets of positive integers whose right half (exclusive) sums to n.

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Author

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Examples

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Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions