A275870 -id:A275870 - cp's OEIS frontend

A303386 Number of aperiodic factorizations of n > 1.

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 2, 4, 1, 5, 1, 6, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 7, 1, 5, 1, 7, 5

Offset: 2

Gus Wiseman, Apr 23 2018

nonn

An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.

			The a(36) = 7 aperiodic factorizations are (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), and (36). Missing from this list are (2*2*3*3) and (6*6).

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A000740, a(2^n) = A000837(n), A001055, A007716, A007916, A045778, A052409, A052410, A100953, A162247, A275024, A275870, A281113, A281116, A301700, A302242, A303431.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],GCD@@Length/@Split[#]===1&]],{n,2,100}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A303386(n) = if(1==n,n,my(r); sumdiv(A052409(n),d, ispower(n,d,&r); moebius(d)*A001055(r))); \\ Antti Karttunen, Sep 25 2018

a(n) = Sum_{d|A052409(n)} mu(d) * A001055(n^(1/d)), where mu = A008683.

More terms from Antti Karttunen, Sep 25 2018

A304465 If n is prime, set a(n) = 1. Otherwise, start with the multiset of prime factors of n, and given a multiset take the multiset of its multiplicities. Repeating this until a multiset of size 1 is reached, set a(n) to the unique element of this multiset.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 6, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 4, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3

Offset: 1

Gus Wiseman, May 13 2018

nonn

a(1) = 0 by convention.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 08 2018

			Starting with the multiset of prime factors of 2520 we have {2,2,2,3,3,5,7} -> {1,1,2,3} -> {1,1,2} -> {1,2} -> {1,1} -> {2}, so a(2520) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001222, A001597, A005117, A007916, A055932, A056239, A112798, A181819, A182850, A182857, A275870, A296150, A303945, A304464.

Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,100}]

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A304465(n) = if(1==n,0,my(t=isprimepower(n)); if(t,t, t=omega(n); if(bigomega(n)==t),t,A304465(A181819(n)))); \\ Antti Karttunen, Nov 08 2018

a(p^n) = n where p is any prime number.
a(product of n distinct primes) = n.
a(1) = 0; and for n > 1, if n = prime^k, a(n) = k, otherwise, if n is squarefree [i.e., A001221(n) = A001222(n)], a(n) = A001221(n), otherwise a(n) = a(A181819(n)). - Antti Karttunen, Nov 08 2018

More terms from Antti Karttunen, Nov 08 2018

A353838 Numbers whose prime indices have all distinct run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, May 23 2022

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 sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

			The prime indices of 180 are {1,1,2,2,3}, with run-sums (2,4,3), so 180 is in the sequence.
The prime indices of 315 are {2,2,3,4}, with run-sums (4,3,4), so 315 is not in the sequence.

The version for all equal run-sums is A353833, counted by A304442.
These partitions are counted by A353837.
The complement is A353839.
The version for compositions is A353852, counted by A353850.
The greatest run-sum is given by A353862, least A353931.
The weak case is A353866, counted by A353864.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A098859 counts partitions with distinct multiplicities, ranked by A130091.
A165413 counts distinct run-sums in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions.
A353832 represents taking run-sums of a partition, compositions A353847.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A002110, A071625, A073093, A118914, A124010, A353834, A353861, A353867.

Select[Range[100],UnsameQ@@Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]&]

A362615 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k co-modes.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 10, 4, 1, 0, 13, 7, 2, 0, 16, 11, 3, 0, 23, 14, 4, 1, 0, 30, 19, 6, 1, 0, 35, 29, 11, 2, 0, 50, 34, 14, 3, 0, 61, 46, 23, 5, 0, 73, 69, 27, 6, 1, 0, 95, 81, 44, 10, 1, 0, 123, 105, 53, 14, 2

Offset: 0

Gus Wiseman, May 04 2023

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

			Triangle begins:
   1
   0   1
   0   2
   0   2   1
   0   4   1
   0   5   2
   0   7   3   1
   0  10   4   1
   0  13   7   2
   0  16  11   3
   0  23  14   4   1
   0  30  19   6   1
   0  35  29  11   2
   0  50  34  14   3
   0  61  46  23   5
   0  73  69  27   6   1
   0  95  81  44  10   1
Row n = 8 counts the following partitions:
  (8)         (53)     (431)
  (44)        (62)     (521)
  (332)       (71)
  (422)       (3221)
  (611)       (3311)
  (2222)      (4211)
  (5111)      (32111)
  (22211)
  (41111)
  (221111)
  (311111)
  (2111111)
  (11111111)

Alois P. Heinz, Rows n = 0..800, flattened

Row sums are A000041.
Row lengths are A002024.
Removing columns 0 and 1 and taking sums gives A362609, ranks A362606.
Column k = 1 is A362610, ranks A359178.
This statistic (co-mode count) is ranked by A362613.
For mode instead of co-mode we have A362614, ranked by A362611.
A008284 counts partitions by length.
A096144 counts partitions by number of minima, A026794 by maxima.
A238342 counts compositions by number of minima, A238341 by maxima.
A275870 counts collapsible partitions.
Cf. A003056, A098859, A325347, A359893, A362607, A362608, A362612, A372632.

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],Length[comsi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

Sum_{k=0..A003056(n)} k * T(n,k) = A372632(n). - Alois P. Heinz, May 07 2024

A362610 Number of integer partitions of n having a unique part of least multiplicity.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 23, 30, 35, 50, 61, 73, 95, 123, 139, 187, 216, 269, 328, 411, 461, 594, 688, 836, 980, 1211, 1357, 1703, 1936, 2330, 2697, 3253, 3649, 4468, 5057, 6005, 6841, 8182, 9149, 10976, 12341, 14508, 16447, 19380, 21611, 25553, 28628

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

Alternatively, these are partitions with a part appearing fewer times than each of the others.

			The partition (3,3,2,2,2,1,1,1) has least multiplicity 2, and only one part of multiplicity 2 (namely 3), so is counted under a(15).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

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

For parts instead of multiplicities we have A002865, ranks A247180.
For median instead of co-mode we have A238478, complement A238479.
These partitions have ranks A359178.
For mode complement of co-mode we have A362607, ranks A362605.
For mode instead of co-mode we have A362608, ranks A356862.
The complement is counted by A362609, ranks A362606.
A000041 counts integer partitions.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A008284, A053263, A098859, A237984, A240219, A304442, A327472, A353863, A353864, A353865, A362612.

Table[Length[Select[IntegerPartitions[n],Count[Length/@Split[#],Min@@Length/@Split[#]]==1&]],{n,0,30}]

seq(n) = my(A=O(x*x^n)); Vec(sum(m=2, n+1, sum(j=1, n, x^(j*(m-1))/(1 + if(j*m<=n, x^(j*m)/(1-x^j) )) + A)*prod(j=1, n\m, 1 + x^(j*m)/(1 - x^j) + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023

G.f.: Sum_{m>=2} (Sum_{j>=1} x^(j*(m-1))/(1 + x^(j*m)/(1 - x^j))) * (Product_{j>=1} (1 + x^(j*m)/(1 - x^j))). - Andrew Howroyd, May 04 2023

A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 46, 59, 60, 63, 64, 127, 128, 136, 138, 143, 168, 170, 175, 187, 238, 248, 250, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047, 2048, 2080, 2084, 2090, 2111, 2184

Offset: 0

Gus Wiseman, May 30 2022

nonn

Every sequence can be uniquely split into non-overlapping runs, read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and corresponding compositions begin:
     0:       0  ()
     1:       1  (1)
     2:      10  (2)
     3:      11  (1,1)
     4:     100  (3)
     7:     111  (1,1,1)
     8:    1000  (4)
    10:    1010  (2,2)
    11:    1011  (2,1,1)
    14:    1110  (1,1,2)
    15:    1111  (1,1,1,1)
    16:   10000  (5)
    31:   11111  (1,1,1,1,1)
    32:  100000  (6)
    36:  100100  (3,3)
    39:  100111  (3,1,1,1)
    42:  101010  (2,2,2)
    46:  101110  (2,1,1,2)
    59:  111011  (1,1,2,1,1)
    60:  111100  (1,1,1,3)
For example:
- The 59th composition in standard order is (1,1,2,1,1), with run-sums (2,2,2), so 59 is in the sequence.
- The 2298th composition in standard order is (4,1,1,1,1,2,2), with run-sums (4,4,4), so 2298 is in the sequence.
- The 2346th composition in standard order is (3,3,2,2,2), with run-sums (6,6), so 2346 is in the sequence.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Standard compositions are listed by A066099.
For equal lengths instead of sums we have A353744, counted by A329738.
The version for partitions is A353833, counted by A304442.
These compositions are counted by A353851.
The distinct instead of equal version is A353852, counted by A353850.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
Cf. A003242, A047966, A106356, A140690, A238279, A274174, A333381, A333489, A333755, A353832, A353864.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],SameQ@@Total/@Split[stc[#]]&]

A353849(a(n)) = 1.

A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 1, 2, 3, 4, 3, 1, 4, 2, 2, 1, 3, 1, 2, 1, 2, 2, 4, 5, 4, 1, 3, 2, 3, 2, 2, 3, 4, 1, 2, 1, 2, 2, 3, 1, 4, 1, 3, 1, 1, 4, 1, 2, 2, 2, 3, 2, 2, 1, 3, 2, 5, 6, 5, 1, 4, 2, 4, 2, 6, 3, 2, 1, 3, 1, 2, 3, 3, 2, 4, 2, 3, 1, 6, 4, 2, 2, 1, 3

Offset: 1

Gus Wiseman, Jun 10 2022

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Triangle begins:
  1
  2
  2
  3
  2 1
  1 2
  3
  4
  3 1
  4
  2 2
  1 3
  1 2 1
For example, composition 350 in standard order is (2,2,1,1,1,2), so row 350 is (4,3,2).

Row-sums are A029837.
Standard compositions are listed by A066099.
Row-lengths are A124767.
These compositions are ranked by A353847.
Row k has A353849(k) distinct parts.
The version for partitions is A354584, ranked by A353832.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A175413, A181819, A238279, A274174, A333381, A333489, A333755, A353835, A353839, A353863, A353864.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total/@Split[stc[n]],{n,0,30}]

A353851 Number of integer compositions of n with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 8, 2, 12, 5, 8, 2, 34, 2, 8, 8, 43, 2, 52, 2, 70, 8, 8, 2, 282, 5, 8, 18, 214, 2, 386, 2, 520, 8, 8, 8, 1957, 2, 8, 8, 2010, 2, 2978, 2, 3094, 94, 8, 2, 16764, 5, 340, 8, 12310, 2, 26514, 8, 27642, 8, 8, 2, 132938, 2, 8, 238, 107411, 8, 236258

Offset: 0

Gus Wiseman, May 31 2022

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The a(0) = 1 through a(8) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                        (112)            (222)                (224)
                        (211)            (1113)               (422)
                        (1111)           (2112)               (2222)
                                         (3111)               (11114)
                                         (11211)              (41111)
                                         (111111)             (111122)
                                                              (112112)
                                                              (211211)
                                                              (221111)
                                                              (11111111)
For example:
  (1,1,2,1,1) has run-sums (2,2,2) so is counted under a(6).
  (4,1,1,1,1,2,2) has run-sums (4,4,4) so is counted under a(12).
  (3,3,2,2,2) has run-sums (6,6) so is counted under a(12).

David A. Corneth, Table of n, a(n) for n = 0..10000

The version for parts or runs instead of run-sums is A000005.
The version for multiplicities instead of run-sums is A098504.
All parts are divisors of n, see A100346.
The version for partitions is A304442, ranked by A353833.
The version for run-lengths instead of run-sums is A329738, ptns A047966.
These compositions are ranked by A353848.
The distinct instead of equal version is A353850.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A353847 represents the composition run-sum transformation.
For distinct instead of equal run-sums: A032020, A098859, A242882, A329739, A351013, A353837, ranked by A353838 (complement A353839), A353852, A354580, ranked by A354581.
Cf. A000005, A006881, A238279, A275870, A333755, A351014, A351016, A351017, A353832, A353834, A353849, A353853-A353859 (run-sum trajectory), A353860, A353863, A353864, A353932.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],SameQ@@Total/@Split[#]&]],{n,0,15}]

a(n) = {if(n <=1, return(1)); my(d = divisors(n), res = 0); for(i = 1, #d, nd = numdiv(d[i]); res+=(nd*(nd-1)^(n/d[i]-1)) ); res } \\ David A. Corneth, Jun 02 2022

From David A. Corneth, Jun 02 2022 (Start)
a(p) = 2 for prime p.
a(p*q) = 8 for distinct primes p and q (Cf. A006881).
a(n) = Sum_{d|n} tau(d)*(tau(d)-1) ^ (n/d - 1) where tau = A000005. (End)

More terms from David A. Corneth, Jun 02 2022

A362607 Number of integer partitions of n with more than one mode.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 4, 6, 9, 13, 13, 23, 23, 33, 45, 56, 64, 90, 101, 137, 169, 208, 246, 320, 379, 469, 567, 702, 828, 1035, 1215, 1488, 1772, 2139, 2533, 3076, 3612, 4333, 5117, 6113, 7168, 8557, 10003, 11862, 13899, 16385, 19109, 22525, 26198, 30729, 35736

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The partition (3,2,2,1,1) has greatest multiplicity 2, and two parts of multiplicity 2 (namely 1 and 2), so is counted under a(9).
The a(3) = 1 through a(9) = 9 partitions:
  (21)  (31)  (32)  (42)    (43)   (53)    (54)
              (41)  (51)    (52)   (62)    (63)
                    (321)   (61)   (71)    (72)
                    (2211)  (421)  (431)   (81)
                                   (521)   (432)
                                   (3311)  (531)
                                           (621)
                                           (32211)
                                           (222111)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 301 terms from John Tyler Rascoe)

For parts instead of multiplicities we have A002865.
For median instead of mode we have A238479, complement A238478.
These partitions have ranks A362605.
The complement is counted by A362608, ranks A356862.
For co-mode we have A362609, ranks A362606.
For co-mode complement we have A362610, ranks A359178.
A000041 counts integer partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A008284, A098859, A237984, A275870, A304442, A327472, A353864, A353865, A360071, A362612.

b:= proc(n, i, m, t) option remember; `if`(n=0, `if`(t=2, 1, 0), `if`(i<1, 0,
      add(b(n-i*j, i-1, max(j, m), `if`(j>m, 1, `if`(j=m, 2, t))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..51);  # Alois P. Heinz, May 05 2024

Table[Length[Select[IntegerPartitions[n],Length[Commonest[#]]>1&]],{n,0,30}]

G_x(N)={my(x='x+O('x^(N-1)), Ib(k,j) = if(k>j,1,0), A_x(u)=sum(i=1,N-u, sum(j=u+1, N-u, (x^(i*(u+j))*(1-x^u)*(1-x^j))/((1-x^(u*i))*(1-x^(j*i))) * prod(k=1,N-i*(u+j), (1-x^(k*(i+Ib(k,j))))/(1-x^k)))));
concat([0,0,0],Vec(sum(u=1,N, A_x(u))))}
G_x(51) \\ John Tyler Rascoe, Apr 05 2024

G.f.: Sum_{u>0} A(u,x) where A(u,x) = Sum_{i>0} Sum_{j>u} ( x^(i*(u+j))*(1-x^u)*(1-x^j) )/( (1-x^(u*i))*(1-x^(j*i)) ) * Product_{k>0} ( (1-x^(k*(i+[k>j])))/(1-x^k) ) is the g.f. for partitions of this kind with least mode u and [] is the Iverson bracket. - John Tyler Rascoe, Apr 05 2024

A362612 Number of integer partitions of n such that the greatest part is the unique mode.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 6, 6, 7, 9, 10, 12, 15, 16, 19, 23, 26, 32, 37, 41, 48, 58, 65, 75, 88, 101, 115, 135, 151, 176, 200, 228, 261, 300, 336, 385, 439, 498, 561, 641, 717, 818, 921, 1036, 1166, 1321, 1477, 1667, 1867, 2099, 2346, 2640, 2944, 3303, 3684

Offset: 0

Gus Wiseman, May 03 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(1) = 1 through a(10) = 7 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    221    33      331      44        333        55
              1111  11111  222     2221     332       441        442
                           111111  1111111  2222      3321       3331
                                            22211     22221      22222
                                            11111111  111111111  222211
                                                                 1111111111

John Tyler Rascoe, Table of n, a(n) for n = 0..500

For median instead of mode we have A053263, complement A237821.
These partitions have ranks A362616.
A000041 counts integer partitions.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362607 counts partitions with more than one mode, ranks A362605.
A362608 counts partitions with a unique mode, ranks A356862.
A362611 counts modes in prime factorization.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A070003, A098859, A102750, A237984, A238478, A238479, A327472, A362609, A362610.

Table[Length[Select[IntegerPartitions[n],Commonest[#]=={Max[#]}&]],{n,0,30}]

A_x(N)={my(x='x+O('x^N), g=sum(i=1, N, sum(j=1, N/i, x^(i*j)*prod(k=1,i-1,(1-x^(j*k))/(1-x^k))))); concat([0],Vec(g))}
A_x(60) \\ John Tyler Rascoe, Apr 03 2024

G.f.: Sum_{i, j>0} x^(i*j) * Product_{k=1,i-1} ((1-x^(j*k))/(1-x^k)). - John Tyler Rascoe, Apr 03 2024

cp's OEIS Frontend

A303386 Number of aperiodic factorizations of n > 1.

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A304465 If n is prime, set a(n) = 1. Otherwise, start with the multiset of prime factors of n, and given a multiset take the multiset of its multiplicities. Repeating this until a multiset of size 1 is reached, set a(n) to the unique element of this multiset.

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A353838 Numbers whose prime indices have all distinct run-sums.

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A362615 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k co-modes.

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A362610 Number of integer partitions of n having a unique part of least multiplicity.

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A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

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A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

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A353851 Number of integer compositions of n with all equal run-sums.