A066328 -id:A066328 - cp's OEIS frontend

A373957 Greatest number of runs in a permutation of the prime factors of n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 1, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 3, 1, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 1, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 3, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Jul 06 2024

nonn

If n belongs to A335433 (the separable case), then a(n) = A001222(n). A multiset is separable iff it has a permutation that is an anti-run (meaning there are no adjacent equal parts).

			The prime factors of 24 are {2,2,2,3}, with permutations (2,2,2,3), (2,2,3,2), (2,3,2,2), (3,2,2,2), with runs:
  ((2,2,2),(3))
  ((2,2),(3),(2))
  ((2),(3),(2,2))
  ((3),(2,2,2))
with lengths (2,3,3,2), with maximum a(24) = 3.

The minimum instead of maximum is A001221.
Positions of 2 are A006881.
Positions of first appearances appear to be A026549.
Positions of 1 are A246655.
The variation A374246 is the difference from bigomega (A001222).
The variation A374247 is the difference with omega (A001221).
This is the last position of a positive term in row n of A374252.
A001221 counts distinct prime factors, A001222 with multiplicity.
A008480 counts permutations of prime factors.
A056239 adds up prime indices, row sums of A112798.
A124767 counts runs in standard compositions, anti-runs A333381.
A304038 is run-compression of prime indices, sums A066328, factors A027748.
A333755 counts compositions by number of runs.
A335433 lists numbers whose prime factors are separable, complement A335448.
Cf. A003242, A007947, A238130, A316413, A373948, A373949, A374250, A374251.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Max@@Table[Length[Split[y]],{y,Permutations[prifacs[n]]}],{n,100}]

a(n) = A374247(n) - A001221(n).

a(n) = A001222(n) - A374246(n).

A381437 Last part of the section-sum partition of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 28 2025

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 section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The prime indices of 972 are {1,1,2,2,2,2,2}, with section-sum partition (3,3,2,2,2), so a(972) = 2.

Positions of first appearances are A008578.
The length of this partition is A051903.
The conjugate version is A051904.
For first instead of last part we get A066328.
These partitions are counted by A239455, complement A351293.
Positions of 1 are A360013, complement A381439.
This is the least prime index of A381431 (see A381432, A381433, A381434, A381435).
This is the last part of row n of A381436 (see A381440, A048767, A351294, A351295).
Counting partitions by this statistic gives A381438.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Cf. A000720, A001221, A001222, A003557, A047966, A116861, A130091, A181819, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[If[n==1,0,Last[egs[prix[n]]]],{n,100}]

a(n) = A055396(A381431(n)).

A381452 Number of multisets that can be obtained by partitioning the prime indices of n into a set of multisets and taking their sums.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A045778 at a(24) = 4, A045778(24) = 5.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into distinct factors > 1.
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.
A multiset partition can be regarded as an arrow in the poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Sets of multisets are generally not transitive. For example, we have arrows: {{1},{2},{1,2}}: {1,1,2,2} -> {1,2,3} and {{1,2},{3}}: {1,2,3} -> {3,3}, but there is no set of multisets {1,1,2,2} -> {3,3}.

			The prime indices of 24 are {1,1,1,2}, with 5 partitions into a set of multisets:
  {{1,1,1,2}}
  {{1},{1,1,2}}
  {{2},{1,1,1}}
  {{1,1},{1,2}}
  {{1},{2},{1,1}}
with block-sums: {5}, {1,4}, {2,3}, {2,3}, {1,2,2}, of which 4 are distinct, so a(24) = 4.

Before taking sums we had A045778.
If each block is a set we have A381441, before sums A050326.
For distinct block-sums instead of blocks we have A381637, before sums A321469.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For sets of constant multisets (A050361) see A381715.
- For set systems with distinct sums (A381633) see A381634, zeros A293243.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
More on sets of multisets: A261049, A317776, A317775, A296118, A318286.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A001970, A002846, A066328, A213385, A213427, A299200, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@#&]]],{n,100}]

a(A002110(n)) = A066723(n).

A360551 Numbers > 1 whose distinct prime indices have non-integer median.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 33, 35, 36, 38, 45, 48, 51, 52, 54, 56, 58, 65, 69, 72, 74, 75, 76, 77, 86, 93, 95, 96, 98, 99, 104, 106, 108, 112, 116, 119, 122, 123, 135, 141, 142, 143, 144, 145, 148, 152, 153, 158, 161, 162, 172, 175, 177, 178, 185, 192, 196

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A325700 in having 330 and lacking 462.
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. Distinct prime indices are listed by A304038.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is not in the sequence.
The prime indices of 462 are {1,2,4,5}, with distinct parts {1,2,4,5}, with median 3, so 462 is not in the sequence.

For mean instead of median we have the complement of A326621.
Positions of odd terms in A360457.
The complement (without 1) is A360550, counted by A360686.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551 complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices, length A001221, sum A066328.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A316413, A348551, A360006, A360009, A360248, A360453.

Select[Range[2,100],!IntegerQ[Median[PrimePi/@First/@FactorInteger[#]]]&]

A365004 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of an integer partition of k.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 4, 4, 1, 0, 7, 7, 8, 4, 1, 0, 11, 12, 17, 13, 6, 1, 0, 15, 19, 30, 28, 18, 6, 1, 0, 22, 30, 53, 58, 50, 24, 8, 1, 0, 30, 45, 86, 109, 108, 70, 33, 8, 1, 0, 42, 67, 139, 194, 223, 179, 107, 40, 10, 1, 0, 56, 97, 213, 328, 420, 394, 286, 143, 50, 10, 1, 0

Offset: 0

Gus Wiseman, Aug 23 2023

A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

			Array begins:
  1  1  2   3   5    7     11
  0  1  2   4   7    12    19
  0  1  4   8   17   30    53
  0  1  4   13  28   58    109
  0  1  6   18  50   108   223
  0  1  6   24  70   179   394
  0  1  8   33  107  286   696
  0  1  8   40  143  428   1108
  0  1  10  50  199  628   1754
  0  1  10  61  254  882   2622
  0  1  12  72  332  1215  3857
  0  1  12  84  410  1624  5457
  0  1  14  99  517  2142  7637
The A(4,2) = 6 ways:
  2*2
  0*1+4*1
  1*1+3*1
  2*1+2*1
  3*1+1*1
  4*1+0*1

Alois P. Heinz, Rows n = 0..200, flattened

Row n = 0 is A000041, strict A000009.
Row n = 1 is A000070.
Column k = 0 is A000007.
Column k = 1 is A000012.
Column k = 2 is A052928 except initial terms.
Antidiagonal sums are A006951.
The case of strict integer partitions is A116861.
Main diagonal is A364907.
The transpose is A364912, also the positive version.
A008284 counts partitions by length, strict A008289.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A066328, A108917, A237113, A364272, A364910, A364911, A364915, A365002.

b:= proc(n, i, m) option remember; `if`(n=0, `if`(m=0, 1, 0),
     `if`(i<1, 0, b(n, i-1, m)+add(b(n-i, min(i, n-i), m-i*j), j=0..m/i)))
    end:
A:= (n, k)-> b(k$2, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jan 28 2024

nn=5;
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
tabv=Table[Length[Join@@Table[combs[n,ptn],{ptn,IntegerPartitions[k]}]],{n,0,nn},{k,0,nn}]
Table[tabv[[k+1,n-k+1]],{n,0,nn},{k,0,n}]

Also the number of ways to write n-k as a *positive* linear combination of an integer partition of k.

Antidiagonals 8-11 from Alois P. Heinz, Jan 28 2024

A367583 Greatest element in row n of A367579 (multiset multiplicity kernel).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 28 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.

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.

			For 450 = 2^1 * 3^2 * 5^2, we have MMK({1,2,2,3,3}) = {1,2,2} so a(450) = 2.

Positions of first appearances are A008578.
Depends only on rootless base A052410, see A007916, A052409.
For minimum instead of maximum element we have A055396.
Row maxima of A367579.
Greatest prime index of A367580.
Positions of 1's are A367586 (powers of even squarefree numbers).
The opposite version is A367587.
A007947 gives squarefree kernel.
A072774 lists powers of squarefree numbers.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reverse A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A363486 gives least prime index of greatest exponent.
A363487 gives greatest prime index of greatest exponent.
A364191 gives least prime index of least exponent.
A364192 gives greatest prime index of least exponent.
Cf. A000720, A000961, A051904, A061395, A071625, A130091, A289023, A367581, A367584, A367585, A367683.

mmk[q_]:=With[{mts=Length/@Split[q]},Sort[Table[Min@@Select[q,Count[q,#]==i&],{i,mts}]]];
Table[If[n==1,0,Max@@mmk[PrimePi/@Join@@ConstantArray@@@If[n==1,{},FactorInteger[n]]]],{n,1,100}]

a(n) = A061395(A367580(n)).
a(n^k) = a(n) for all positive integers n and k.
If n is a power of a squarefree number, a(n) = A055396(n).

A380956 Position of first appearance of n in A380955 (sum of prime indices minus sum of distinct prime indices).

Original entry on oeis.org

1, 4, 8, 16, 27, 64, 81, 256, 243, 529, 729, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 0

Gus Wiseman, Feb 12 2025

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.

Also the position of first appearance of n in A374248.

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   243: {2,2,2,2,2}
   529: {9,9}
   729: {2,2,2,2,2,2}
   961: {11,11}
  1369: {12,12}
  1681: {13,13}
  1849: {14,14}
  2209: {15,15}

For length instead of sum we have A151821.
For factors instead of indices we have A280286 (sorted A381075), firsts of A280292.
Counting partitions by this statistic gives A364916.
Positions of first appearances in A380955.
The sorted version is A380957.
For product instead of sum we have firsts of A380986.
A multiplicative version is A380987 (sorted A380988), firsts of A290106.
For prime multiplicities instead of prime indices we have A380989, firsts of A380958.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A325033, A366528, A366749, A374248.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Table[Total[prix[n]]-Total[Union[prix[n]]],{n,1000}];
Table[Position[q,k][[1,1]],{k,0,mnrm[q+1]-1}]

After a(12) = 961, this appears to converge to prime(n)^2.

A367582 Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 1, 4, 3, 3, 2, 1, 0, 1, 3, 5, 4, 4, 3, 1, 1, 0, 1, 2, 6, 4, 8, 3, 3, 2, 1, 0, 1, 3, 7, 9, 6, 7, 4, 3, 1, 1, 0, 1, 1, 8, 7, 11, 9, 9, 4, 3, 2, 1

Offset: 0

Gus Wiseman, Nov 28 2023

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  1  1
  0  1  1  2  2  1
  0  1  3  3  2  1  1
  0  1  1  4  3  3  2  1
  0  1  3  5  4  4  3  1  1
  0  1  2  6  4  8  3  3  2  1
  0  1  3  7  9  6  7  4  3  1  1
  0  1  1  8  7 11  9  9  4  3  2  1
  0  1  5 10 11 13 10 11  6  5  3  1  1
  0  1  1 10 11 17 14 18 10  9  4  3  2  1
  0  1  3 12 17 19 18 22 14 12  8  4  3  1  1
  0  1  3 12 15 27 19 31 19 19 10  9  5  3  2  1
  0  1  4 15 23 27 31 33 24 26 18 12  8  4  3  1  1
  0  1  1 14 20 35 33 48 32 37 25 20 11 10  4  3  2  1
Row n = 7 counts the following partitions:
  (1111111)  (61)  (421)     (52)     (4111)  (511)  (7)
                   (2221)    (331)    (322)   (43)
                   (22111)   (31111)  (3211)
                   (211111)

Column k = 2 is A000005(n) - 1 = A032741(n).
Row sums are A000041.
The case of constant partitions is A051731, row sums A000005.
The corresponding rank statistic is A367581, row sums of A367579.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
A116861 counts partitions by sum of distinct parts.
Cf. A000837, A056239, A066328, A071625, A072774, A082090, A367580.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], Total[mmk[#]]==k&]], {n,0,10}, {k,0,n}]

A381435 Numbers appearing more than once in A381431 (section-sum partition of prime indices).

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 25, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 49, 51, 52, 53, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 101, 103, 104, 106, 107, 109, 111, 113, 115, 116, 117, 118, 119

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
   5: {3}
   7: {4}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  26: {1,6}
  29: {10}
  31: {11}
  34: {1,7}
  37: {12}
  38: {1,8}
  39: {2,6}
  41: {13}
  43: {14}
  46: {1,9}
  47: {15}
  49: {4,4}
  51: {2,7}
  52: {1,1,6}

In A381431:
- fixed points are A000961, A000005
- conjugate is A048767, fixed points A048768, A217605
- all numbers present are A381432, conjugate A351294
- numbers missing are A381433, conjugate A351295
- numbers appearing only once are A381434, conjugate A381540
- numbers appearing more than once are A381435 (this), conjugate A381541
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions, complement A351293.
A381436 lists section-sum partition of prime indices, conjugate A381440.
Set multipartitions: A050320, A089259, A116540, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A212166, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],Count[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]>1&]

The complement is A381434 U A381433.

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 2, 3, 7, 0, 11, 5, 6, 0, 15, 2, 22, 0, 10, 7, 30, 0, 6, 11, 0, 0, 42, 6, 56, 0, 14, 15, 15, 0, 77, 22, 22, 0, 101, 10, 135, 0, 6, 30, 176, 0, 20, 6, 30, 0, 231, 0, 21, 0, 44, 42, 297, 0, 385, 56, 10, 0, 33, 14, 490, 0, 60, 15, 627, 0

Offset: 1

Gus Wiseman, Aug 18 2025

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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
  ((2),(1,1))
  ((1,1),(2))
The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
  ((2),(3))
  ((2),(2,1))
  ((2),(1,1,1))
  ((1,1),(2,1))
  ((1,1),(1,1,1))

Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).
Allowing repeated partitions gives A299200, A357977, A357982, A357978.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.
For prime factors instead of partitions see A355741, A355742, A387136.
The disjoint case is A383706.
For initial intervals instead of partitions we have A387111.
The case of strict partitions is A387115.
The case of constant partitions is A387120.
Taking each prime factor (instead of index) gives A387133.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A008284, A063834, A261049, A296122, A299201, A335433, A335448, A355745, A387135.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[IntegerPartitions/@prix[n]],UnsameQ@@#&]],{n,100}]

cp's OEIS Frontend

A373957 Greatest number of runs in a permutation of the prime factors of n.

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A381437 Last part of the section-sum partition of the prime indices of n.

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A381452 Number of multisets that can be obtained by partitioning the prime indices of n into a set of multisets and taking their sums.

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A360551 Numbers > 1 whose distinct prime indices have non-integer median.

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A365004 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of an integer partition of k.

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A367583 Greatest element in row n of A367579 (multiset multiplicity kernel).

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A380956 Position of first appearance of n in A380955 (sum of prime indices minus sum of distinct prime indices).

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A367582 Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.

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