A347044 -id:A347044 - 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}]

A345957 Number of divisors of n with exactly half as many prime factors as n, counting multiplicity.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 16 2021

nonn

These divisors do not necessarily include the central divisors (A207375), and may not themselves be central.

			The a(n) divisors for selected n:
  n = 1:  6:  36:  60:  210:  840:  900:  1260:  1296:  3600:
     --------------------------------------------------------
      1   2    4    4     6     8    12     12     16     16
          3    6    6    10    12    18     18     24     24
               9   10    14    20    20     20     36     36
                   15    15    28    30     28     54     40
                         21    30    45     30     81     60
                         35    42    50     42            90
                               70    75     45           100
                              105           63           150
                                            70           225
                                           105

The case of powers of 2 is A000035.
Positions of even terms are A000037.
Positions of odd terms are A000290.
Positions of 0's are A026424.
Positions of 1's are A056798.
The rounded version is A096825.
The case of all divisors (not just 2) is A347042.
The smallest of these divisors is A347045 (rounded: A347043).
The greatest of these divisors is A347046 (rounded: A347044).
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A334997 counts chains of divisors of n by length.
Cf. A001227, A001414, A028260, A033676, A033677, A073093, A074206, A217581, A344653, A346697-A346704.

Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeOmega[n]/2&]],{n,100}]

a(n) = my(nb=bigomega(n)); sumdiv(n, d, 2*bigomega(d) == nb); \\ Michel Marcus, Aug 16 2021

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    divs = divisors(n)
    return sum(2*len(factorint(d, multiple=True)) == npf for d in divs)
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Aug 17 2021
(Python 3.8+)
from itertools import combinations
from math import prod, comb
from sympy import factorint
def A345957(n):
    if n == 1:
        return 1
    fs = factorint(n)
    elist = list(fs.values())
    q, r = divmod(sum(elist),2)
    k = len(elist)
    if r:
        return 0
    c = 0
    for i in range(k+1):
        m = (-1)**i
        for d in combinations(range(k),i):
            t = k+q-sum(elist[j] for j in d)-i-1
            if t >= 0:
                c += m*comb(t,k-1)
    return c # Chai Wah Wu, Aug 20 2021

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A345957(n):
    if n == 1:
        return 1
    fs = factorint(n,multiple=True)
    q, r = divmod(len(fs),2)
    return 0 if r else len(list(multiset_combinations(fs,q))) # Chai Wah Wu, Aug 20 2021

A347043 Smallest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 4, 13, 2, 3, 4, 17, 6, 19, 4, 3, 2, 23, 4, 5, 2, 9, 4, 29, 6, 31, 8, 3, 2, 5, 4, 37, 2, 3, 4, 41, 6, 43, 4, 9, 2, 47, 8, 7, 10, 3, 4, 53, 6, 5, 4, 3, 2, 59, 4, 61, 2, 9, 8, 5, 6, 67, 4, 3, 10, 71, 8, 73, 2, 15, 4, 7, 6, 79, 8

Offset: 1

Gus Wiseman, Aug 15 2021

nonn

Appears to contain every positive integer at least once.

This is correct. For any integer m, let p be any prime > m. Then a(m*p^A001222(m)) = m. - Sebastian Karlsson, Oct 11 2022

			The divisors of 123456 with half bigomega are: 16, 24, 5144, 7716, so a(123456) = 16.

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

Positions of 2's are A001747.
Positions of odd terms are A005408.
Positions of even terms are A005843.
The case of powers of 2 is A016116.
The smallest divisor without the condition is A020639 (greatest: A006530).
These divisors are counted by A096825 (exact: A345957).
The greatest of these divisors is A347044 (exact: A347046).
The exact version is A347045.
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors (min: A033676, max: A033677).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Cf. A000720, A027746, A038548, A106529, A140271, A244991, A324522, A335433, A335448.

Table[Min[Select[Divisors[n],PrimeOmega[#]==Ceiling[PrimeOmega[n]/2]&]],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]}, Times @@ p[[1 ;; Ceiling[Length[p]/2]]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

a(n) = my(bn=ceil(bigomega(n)/2)); fordiv(n, d, if (bigomega(d)==bn, return (d))); \\ Michel Marcus, Aug 18 2021

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n):
        if len(factorint(d, multiple=True)) == (npf+1)//2: return d
    return 1
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Aug 18 2021

from math import prod
from sympy import factorint
def A347043(n):
    fs = factorint(n,multiple=True)
    l = len(fs)
    return prod(fs[:(l+1)//2]) # Chai Wah Wu, Aug 20 2021

a(n) = Product_{k=1..ceiling(A001222(n)/2)} A027746(n,k). - Amiram Eldar, Nov 02 2024

A360671 Number of multisets whose right half (inclusive) sums to n.

Original entry on oeis.org

1, 2, 5, 8, 16, 21, 42, 51, 90, 121, 185, 235, 386, 465, 679, 908, 1261, 1580, 2238, 2770, 3827, 4831, 6314, 7910, 10619, 13074, 16813, 21049, 26934, 33072, 42445, 51679, 65264, 79902, 99309, 121548, 151325, 182697, 224873, 272625, 334536, 401999, 491560, 588723

Offset: 0

Gus Wiseman, Mar 09 2023

nonn

			The a(0) = 1 through a(4) = 16 multisets:
  {}  {1}    {2}        {3}            {4}
      {1,1}  {1,2}      {1,3}          {1,4}
             {2,2}      {2,3}          {2,4}
             {1,1,1}    {3,3}          {3,4}
             {1,1,1,1}  {1,1,2}        {4,4}
                        {1,1,1,2}      {1,1,3}
                        {1,1,1,1,1}    {1,2,2}
                        {1,1,1,1,1,1}  {2,2,2}
                                       {1,1,1,3}
                                       {1,1,2,2}
                                       {1,2,2,2}
                                       {2,2,2,2}
                                       {1,1,1,1,2}
                                       {1,1,1,1,1,2}
                                       {1,1,1,1,1,1,1}
                                       {1,1,1,1,1,1,1,1}
For example, the multiset y = {1,1,1,1,2} has right half (inclusive) {1,1,2}, with sum 4, so y is counted under a(4).

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

The exclusive version is A360673.
Column sums of A360675 with rows reversed.
The case of sets is A360955, exclusive A360954.
The even-length case is A360956.
A360672 counts partitions by left sum (exclusive).
A360679 gives right sum (inclusive) of prime indices.
Cf. A000041, A347044, A361200, A361201.

Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], Total[Take[#,Ceiling[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+1); 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+1) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023

Terms a(24) and beyond from Andrew Howroyd, Mar 11 2023

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

A360679 Sum of the right half (inclusive) of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 2, 2, 3, 5, 3, 6, 4, 3, 2, 7, 4, 8, 4, 4, 5, 9, 3, 3, 6, 4, 5, 10, 5, 11, 3, 5, 7, 4, 4, 12, 8, 6, 4, 13, 6, 14, 6, 5, 9, 15, 4, 4, 6, 7, 7, 16, 4, 5, 5, 8, 10, 17, 5, 18, 11, 6, 3, 6, 7, 19, 8, 9, 7, 20, 5, 21, 12, 6, 9, 5, 8, 22, 5, 4

Offset: 1

Gus Wiseman, Mar 05 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 right half (inclusive) {2,2,3}, so a(810) = 7.
The prime indices of 3675 are {2,3,3,4,4}, with right half (inclusive) {3,4,4}, so a(3675) = 11.

Positions of first appearances are 1 and A001248.
The value k appears A360671(k) times, exclusive A360673.
These partitions are counted by A360672 with rows reversed.
The exclusive version is A360677.
The left version is A360678.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360616 gives half of bigomega (exclusive), inclusive A360617.
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, A359912, A360006, A360007, A360457.

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

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

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

A361200 Product of the left half (exclusive) of the multiset of prime factors of n; a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 2, 1, 4, 5, 2, 3, 2, 1, 2, 1, 4, 3, 2, 5, 4, 1, 2, 3, 4, 1, 2, 1, 2, 3, 2, 1, 4, 7, 2, 3, 2, 1, 6, 5, 4, 3, 2, 1, 4, 1, 2, 3, 8, 5, 2, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 7, 2, 1, 4, 9, 2, 1, 4, 5, 2, 3

Offset: 1

Gus Wiseman, Mar 10 2023

nonn

			The prime factors of 250 are {2,5,5,5}, with left half (exclusive) {2,5}, with product 10, so a(250) = 10.

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

Positions of 1's are A000040.
Positions of 2's are A037143.
The inclusive version is A347043.
The right inclusive version A347044.
The right version is A361201.
A000005 counts divisors.
A001221 counts distinct prime factors.
A006530 gives greatest prime factor.
A112798 lists prime indices, length A001222, sum A056239.
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. A001248, A026424, A027746, A096825, A347045, A347046, A360005.

Table[If[n==1,0,Times@@Take[Join@@ConstantArray@@@FactorInteger[n],Floor[PrimeOmega[n]/2]]],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]}, Times @@ p[[1 ;; Floor[Length[p]/2]]]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

a(n) * A347044(n) = n.
A361201(n) * A347043(n) = n.
a(n) = Product_{k=1..floor(A001222(n)/2)} A027746(n,k) for n >= 2. - Amiram Eldar, Nov 02 2024

A361201 Product of the right half (exclusive) of the multiset of prime factors of n; a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 3, 1, 7, 5, 4, 1, 3, 1, 5, 7, 11, 1, 6, 5, 13, 3, 7, 1, 5, 1, 4, 11, 17, 7, 9, 1, 19, 13, 10, 1, 7, 1, 11, 5, 23, 1, 6, 7, 5, 17, 13, 1, 9, 11, 14, 19, 29, 1, 15, 1, 31, 7, 8, 13, 11, 1, 17, 23, 7, 1, 9, 1, 37, 5, 19, 11, 13, 1

Offset: 1

Gus Wiseman, Mar 10 2023

			The prime factors of 250 are {2,5,5,5}, with right half (exclusive) {5,5}, with product 25, so a(250) = 25.

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

Positions of 1's are A000040.
Positions of first appearances are A123666.
The left inclusive version A347043.
The inclusive version is A347044.
The left version is A361200.
A000005 counts divisors.
A001221 counts distinct prime factors.
A006530 gives greatest prime factor.
A112798 lists prime indices, length A001222, sum A056239.
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. A001248, A026424, A096825, A347045, A347046, A360005.

f:= proc(n) local F;
  F:= ifactors(n)[2];
  F:= sort(map(t -> t[1]$t[2],F));
  convert(F[ceil(nops(F)/2)+1 ..-1],`*`)
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Aug 12 2024

Table[If[n==1,0,Times@@Take[Join@@ConstantArray@@@FactorInteger[n],-Floor[PrimeOmega[n]/2]]],{n,100}]

A361200(n) * A347044(n) = n.

A361201(n) * A347043(n) = n.

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).

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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A345957 Number of divisors of n with exactly half as many prime factors as n, counting multiplicity.

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A347043 Smallest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.

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A360671 Number of multisets whose right half (inclusive) sums to n.

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

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A360679 Sum of the right half (inclusive) of the prime indices of n.

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A361200 Product of the left half (exclusive) of the multiset of prime factors of n; a(1) = 0.