A047966 -id:A047966 - cp's OEIS frontend

A097986 Number of strict integer partitions of n with a part dividing all the other parts.

1, 1, 2, 2, 2, 4, 3, 5, 5, 7, 6, 12, 9, 13, 15, 20, 18, 28, 26, 37, 39, 47, 49, 71, 68, 85, 94, 117, 120, 159, 160, 201, 216, 257, 277, 348, 357, 430, 470, 562, 592, 720, 758, 901, 981, 1134, 1220, 1457, 1542, 1798, 1952, 2250, 2419, 2819, 3023, 3482, 3773, 4291

Offset: 1

Vladeta Jovovic, Oct 23 2004

If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 23 2021

Also the number of uniform (constant multiplicity) partitions of n containing 1, ranked by A367586. The strict case is A096765. The version without 1 is A329436. - Gus Wiseman, Dec 01 2023

			From _Gus Wiseman_, Dec 01 2023: (Start)
The a(1) = 1 through a(8) = 5 strict partitions with a part dividing all the other parts:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (2,1)  (3,1)  (4,1)  (4,2)    (6,1)    (6,2)
                                 (5,1)    (4,2,1)  (7,1)
                                 (3,2,1)           (4,3,1)
                                                   (5,2,1)
The a(1) = 1 through a(8) = 5 uniform partitions containing 1:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (71)
             (111)  (1111)  (11111)  (321)     (421)      (431)
                                     (2211)    (1111111)  (521)
                                     (111111)             (3311)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 500 terms from John Tyler Rascoe)

The non-strict version is A083710.
The case with no 1's is A098965.
The Heinz numbers of these partitions are A339563.
The strict complement is counted by A341450.
The version for "divisible by" instead of "dividing" is A343347.
The case where there is also a part divisible by all the others is A343378.
The case where there is no part divisible by all the others is A343381.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A083711, A098743, A130689, A200745, A264401, A338470, A343377, A343379, A343380.
Cf. A023645, A025147, A047966, A072774, A096765, A329436, A367586.

Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 62}], {k, 62}]], x], {2, 60}] (* Robert G. Wilson v, Nov 01 2004 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Or@@Table[And@@IntegerQ/@(#/x), {x,#}]&]], {n,0,30}] (* Gus Wiseman, Apr 23 2021 *)

A_x(N) = {my(x='x+O('x^N)); Vec(sum(k=1,N,x^k*prod(i=2,N-k, (1+x^(k*i)))))}
A_x(50) \\ John Tyler Rascoe, Nov 19 2024

a(n) = Sum_{d|n} A025147(d-1).
G.f.: Sum_{k>=1} (x^k*Product_{i>=2} (1+x^(k*i))).
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jul 06 2025

More terms from Robert G. Wilson v, Nov 01 2004

Name shortened by Gus Wiseman, Apr 23 2021

A381717 Number of integer partitions of n that cannot be partitioned into constant multisets with distinct block-sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 3, 2, 3, 6, 7, 10, 15, 15, 28, 37, 47, 64, 71, 97, 139, 173, 215, 273, 361, 439, 551, 691, 853, 1078, 1325, 1623, 2046, 2458, 2998, 3697, 4527, 5472, 6590, 7988, 9590, 11598, 13933, 16560, 19976, 23822, 28420, 33797, 40088, 47476, 56369, 66678

Offset: 0

Gus Wiseman, Mar 16 2025

nonn

Conjecture: Also the number of integer partitions of n having no permutation with all distinct run-sums, ranked by zeros of A382876. In other words, a partition has a permutation with all distinct run-sums iff it has a multiset partition into constant blocks with all distinct block-sums, where the run-sums of a sequence are obtained by splitting it into maximal runs and taking their sums.

			For y = (3,2,2,1) we have the multiset partition {{3},{2,2},{1}}, so y is not counted under a(8).
For y = (3,2,1,1,1) there are 3 multiset partitions into constant multisets:
  {{3},{2},{1,1,1}}
  {{3},{2},{1,1},{1}}
  {{3},{2},{1},{1},{1}}
but none of these has distinct block-sums, so y is counted under a(8).
For y = (3,3,1,1,1,1,1,1) we have multiset partitions:
  {{1},{3,3},{1,1,1,1,1}}
  {{1,1},{3,3},{1,1,1,1}}
  {{1},{1,1},{3,3},{1,1,1}}
so y is not counted under a(12).
The a(4) = 1 through a(13) = 10 partitions:
  211  .  .  3211  422    4221  6211   4322     633      5422
                   4211   5211  33211  7211     8211     6331
                   32111        42211  43211    43221    9211
                                       422111   44211    54211
                                       431111   53211    63211
                                       3221111  432111   333211
                                                4221111  432211
                                                         532111
                                                         4321111
                                                         42211111

Twice-partitions of this type (constant with distinct) are counted by A279786.
Multiset partitions of this type are ranked by A326535 /\ A355743.
These partitions are ranked by A381636, zeros of A381635.
For strict instead of constant blocks we have A381990, see A381806, A381633, A382079.
For equal instead of distinct block-sums we have A381993.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383.
A050361 counts factorizations into distinct prime powers.
Cf. A002846, A047966, A130091, A213242, A213427, A265947, A317142, A353864, A381716, A381991, A381992, A382876.

mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Select[IntegerPartitions[n],Select[Join@@@Tuples[mce/@Split[#]],UnsameQ@@Total/@#&]=={}&]],{n,0,30}]

a(37)-a(53) from Robert Price, Mar 31 2025

A382879 Positions of 0 in A382857 (permutations of prime indices with equal run-lengths).

Original entry on oeis.org

24, 40, 48, 54, 56, 80, 88, 96, 104, 112, 135, 136, 152, 160, 162, 176, 184, 189, 192, 208, 224, 232, 240, 248, 250, 272, 288, 296, 297, 304, 320, 328, 336, 344, 351, 352, 368, 375, 376, 384, 405, 416, 424, 448, 459, 464, 472, 480, 486, 488, 496, 513, 528, 536

Offset: 1

Gus Wiseman, Apr 09 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, sum A056239.

			The terms together with their prime indices begin:
   24: {1,1,1,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   80: {1,1,1,1,3}
   88: {1,1,1,5}
   96: {1,1,1,1,1,2}
  104: {1,1,1,6}
  112: {1,1,1,1,4}
  135: {2,2,2,3}
  136: {1,1,1,7}
  152: {1,1,1,8}
  160: {1,1,1,1,1,3}

For distinct instead of equal the complement is A351294, counted by A239455.
For distinct instead of equal we have A351295, counted by A351293.
For run-sums instead of run-lengths we have A383100, zeros of A382877, distinct A382876.
Positions of 0 in A382857 (firsts A382878), by signature A382858 (distinct A382773).
For prime signature instead of prime indices we have A382914.
Partitions of this type are counted by A382915.
The complement is counted by A383013.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798.
A297770 counts distinct runs in binary expansion.
A164707 lists numbers whose binary form has equal runs of ones, distinct A328592.
A304442 counts partitions with equal run-sums, ranks A353833.
A329739 counts compositions with distinct run-lengths, ranks A351290.
A353744 ranks compositions with equal run-lengths, distinct A351596 (complement A351291).
Cf. A000720, A000961, A001221, A001222, A003242, A008480, A047966, A130091, A238279, A351201.

Select[Range[100], Select[Permutations[Join@@ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]=={}&]

A100471 Number of integer partitions of n whose sequence of frequencies is strictly increasing.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 18, 20, 27, 32, 40, 44, 60, 67, 82, 93, 114, 129, 161, 175, 209, 239, 285, 315, 372, 416, 484, 545, 631, 698, 811, 890, 1027, 1146, 1304, 1437, 1631, 1805, 2042, 2252, 2539, 2785, 3143, 3439, 3846, 4226, 4722, 5159

Offset: 0

David S. Newman, Nov 21 2004

nonn

			a(4) = 4 because of the 5 unrestricted partitions of 4, only one, 3+1 uses each of its summands just once and 1,1 is not an increasing sequence.
From _Gus Wiseman_, Jan 23 2019: (Start)
The a(1) = 1 through a(8) = 11 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (4111)     (611)
                                     (3111)    (22111)    (2222)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..8000 (terms 0..1000 from Alois P. Heinz)

Cf. A000219, A000837 (frequencies are relatively prime), A047966 (frequencies are equal), A098859 (frequencies are distinct), A100881, A100882, A100883, A304686 (Heinz numbers of these partitions).

a100471 n = p 0 (n + 1) 1 n where
   p m m' k x | x == 0    = if m < m' || m == 0 then 1 else 0
              | x < k     = 0
              | m == 0    = p 1 m' k (x - k) + p 0 m' (k + 1) x
              | otherwise = p (m + 1) m' k (x - k) +
                            if m < m' then p 0 m (k + 1) x else 0
-- Reinhard Zumkeller, Dec 27 2012

b:= proc(n,i,t) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=1 then `if`(n>t, 1, 0)
    elif i=0 then 0
    else      b(n, i-1, t)
         +add(b(n-i*j, i-1, j), j=t+1..floor(n/i))
      fi
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 21 2011

b[n_, i_, t_] := b[n, i, t] = Which[n<0, 0, n==0, 1, i==1, If[n>t, 1, 0], i == 0, 0 , True, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t+1, Floor[n/i]}]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ@*Split]],{n,20}] (* Gus Wiseman, Jan 23 2019 *)

Corrected and extended by Vladeta Jovovic, Nov 24 2004

Name edited by Gus Wiseman, Jan 23 2019

A381992 Number of integer partitions of n that can be partitioned into sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 77, 100, 134, 170, 217, 282, 360, 449, 571, 719, 899, 1122, 1391, 1727, 2136, 2616, 3209, 3947, 4800, 5845, 7094, 8602, 10408, 12533, 15062, 18107, 21686, 25956, 30967, 36936, 43897, 52132, 61850, 73157, 86466, 101992, 120195

Offset: 0

Gus Wiseman, Mar 16 2025

nonn

Also the number of integer partitions of n whose Heinz number belongs to A382075 (can be written as a product of squarefree numbers with distinct sums of prime indices).

			There are 6 ways to partition (3,2,2,1) into sets:
  {{2},{1,2,3}}
  {{1,2},{2,3}}
  {{1},{2},{2,3}}
  {{2},{2},{1,3}}
  {{2},{3},{1,2}}
  {{1},{2},{2},{3}}
Of these, 3 have distinct block sums:
  {{2},{1,2,3}}
  {{1,2},{2,3}}
  {{1},{2},{2,3}}
so (3,2,2,1) is counted under a(8).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)  (3)    (4)      (5)      (6)        (7)        (8)
            (2,1)  (3,1)    (3,2)    (4,2)      (4,3)      (5,3)
                   (2,1,1)  (4,1)    (5,1)      (5,2)      (6,2)
                            (2,2,1)  (3,2,1)    (6,1)      (7,1)
                            (3,1,1)  (4,1,1)    (3,2,2)    (3,3,2)
                                     (2,2,1,1)  (3,3,1)    (4,2,2)
                                                (4,2,1)    (4,3,1)
                                                (5,1,1)    (5,2,1)
                                                (3,2,1,1)  (6,1,1)
                                                           (3,2,2,1)
                                                           (3,3,1,1)
                                                           (4,2,1,1)
                                                           (3,2,1,1,1)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279785.
Multiset partitions of this type are counted by A381633, zeros of A381634.
For constant instead of strict blocks see A381717, A381636, A381635, A381716, A381991.
Normal multiset partitions of this type are counted by A381718, see A116539.
The complement is counted by A381990, ranked by A381806.
These partitions are ranked by A382075.
For distinct blocks instead of sums we have A382077, complement A382078.
For a unique choice we have A382079.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
A382201 lists MM-numbers of sets with distinct sums.
Cf. A002846, A047966, A213427, A279786, A299202, A317142, A381870.
Cf. A293243, A293511, A358914, A381078, A381441, A381454.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#], And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,10}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 3, 9, 9, 12, 12, 20, 18, 28, 27, 37, 42, 55, 51, 74, 80, 98, 105, 136, 137, 180, 189, 232, 255, 308, 320, 403, 434, 512, 551, 668, 706, 852, 915, 1067, 1170, 1370, 1453, 1722, 1860, 2145, 2332, 2701, 2899, 3355, 3626, 4144

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Number of strict integer partitions of n with no part dividing all the others.

			The a(0) = 1 through a(15) = 12 strict partitions (empty columns indicated by dots, 0 represents the empty partition, A..D = 10..13):
  0  .  .  .  .  32   .  43   53   54    64    65    75    76    86     87
                         52        72    73    74    543   85    95     96
                                   432   532   83    732   94    A4     B4
                                               92          A3    B3     D2
                                               542         B2    653    654
                                               632         643   743    753
                                                           652   752    762
                                                           742   932    843
                                                           832   5432   852
                                                                        942
                                                                        A32
                                                                        6432

The complement is counted by A097986 (non-strict: A083710, rank: A339563).
The complement with no 1's is A098965 (non-strict: A083711).
The non-strict version is A338470.
The Heinz numbers of these partitions are A339562 (non-strict: A342193).
The case with greatest part not divisible by all others is A343379.
The case with greatest part divisible by all others is A343380.
A000009 counts strict partitions (non-strict: A000041).
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001787, A001792, A064410, A264401, A343342, A343381, A343382.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]

a(n > 0) = A000009(n) - Sum_{d|n} A025147(d-1).

A353834 Nonprime numbers whose prime indices have all equal run-sums.

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 25, 27, 32, 40, 49, 63, 64, 81, 112, 121, 125, 128, 144, 169, 243, 256, 289, 325, 343, 351, 352, 361, 512, 529, 625, 675, 729, 832, 841, 931, 961, 1008, 1024, 1331, 1369, 1539, 1600, 1681, 1728, 1849, 2048, 2176, 2187, 2197, 2209, 2401

Offset: 1

Gus Wiseman, May 26 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 terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    12: {1,1,2}
    16: {1,1,1,1}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    40: {1,1,1,3}
    49: {4,4}
    63: {2,2,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   112: {1,1,1,1,4}
   121: {5,5}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}
For example, 675 is in the sequence because its prime indices {2,2,2,3,3} have run-sums (6,6).

For equal run-lengths we have A072774\A000040, counted by A047966(n)-1.
These partitions are counted by A304442(n) - 1.
These are the nonprime positions of prime powers in A353832.
Including the primes gives A353833.
For distinct run-sums we have A353838\A000040, counted by A353837(n)-1.
For compositions we have A353848\A000079, counted by A353851(n)-1.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion, distinct run-lengths A165413.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353840-A353846 pertain to partition run-sum trajectory.
A353862 gives greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A000005, A000961, A007947, A025475, A071625, A073093, A323014, A353839.

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

from itertools import count, islice
from sympy import factorint, primepi
def A353848_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: n == 1 or (sum((f:=factorint(n)).values()) > 1 and len(set(primepi(p)*e for p, e in f.items())) <= 1), count(max(startvalue,1)))
A353848_list = list(islice(A353848_gen(),30)) # Chai Wah Wu, May 27 2022

A338470 Number of integer partitions of n with no part dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 13, 7, 23, 21, 33, 35, 65, 55, 104, 97, 151, 166, 252, 235, 377, 399, 549, 591, 846, 858, 1237, 1311, 1749, 1934, 2556, 2705, 3659, 3991, 5090, 5608, 7244, 7841, 10086, 11075, 13794, 15420, 19195, 21003, 26240, 29089, 35483

Offset: 0

Gus Wiseman, Mar 23 2021

nonn

Alternative name: Number of integer partitions of n that are empty or have smallest part not dividing all the others.

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
           (52)   (332)  (72)    (73)    (74)     (543)
           (322)         (432)   (433)   (83)     (552)
                         (522)   (532)   (92)     (732)
                         (3222)  (3322)  (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

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

The complement is A083710 (strict: A097986).
The strict case is A341450.
The Heinz numbers of these partitions are A342193.
The dual version is A343341.
The case with maximum part not divisible by all the others is A343342.
The case with maximum part divisible by all the others is A343344.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A001787 count normal multisets with a selected position.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
A276024 counts positive subset sums.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001792, A064391, A064410, A066186, A067824, A083711, A098965, A264401, A339562, A339563.

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]
(* Second program: *)
a[n_] := If[n == 0, 1, PartitionsP[n] - Sum[PartitionsP[d-1], {d, Divisors[n]}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 09 2021, after Andrew Howroyd *)

a(n)={numbpart(n) - if(n, sumdiv(n, d, numbpart(d-1)))} \\ Andrew Howroyd, Mar 25 2021

a(n) = A000041(n) - Sum_{d|n} A000041(d-1) for n > 0. - Andrew Howroyd, Mar 25 2021

A101509 Binomial transform of tau(n) (see A000005).

Original entry on oeis.org

1, 3, 7, 16, 35, 75, 159, 334, 696, 1442, 2976, 6123, 12562, 25706, 52492, 107014, 217877, 443061, 899957, 1826078, 3701783, 7498261, 15178255, 30706320, 62085915, 125465715, 253415981, 511608490, 1032427637, 2082680887, 4199956101, 8467124805, 17064784905, 34382825363, 69256687719, 139465867773

Offset: 0

Paul Barry, Dec 05 2004

Row sums of A101508.

Also: Number of matrices with positive integer coefficients such that the sum of all entries equals n+1, cf. link "Partitions and A101509". - M. F. Hasler, Jan 14 2009

			From _Gus Wiseman_, Jan 16 2019: (Start)
The a(3) = 16 ways to arrange the parts of an integer partition of 4 into a matrix:
  [4] [1 3] [3 1] [2 2] [1 1 2] [1 2 1] [2 1 1] [1 1 1 1]
.
  [1] [3] [2] [1 1]
  [3] [1] [2] [1 1]
.
  [1] [1] [2]
  [1] [2] [1]
  [2] [1] [1]
.
  [1]
  [1]
  [1]
  [1]
(End)

M. F. Hasler, Table of n, a(n) for n = 0..500
L. Manor, M. F. Hasler, Partitions and A101509. SeqFan list, Jan 14 2009
N. J. A. Sloane, Transforms

Cf. A000005 (tau), A101508, A160399.

Cf. A000219, A047966, A053529, A319066, A323300, A323301, A323307, A323429.

bintr:= proc(p) proc(n) add(p(k) *binomial(n, k), k=0..n) end end:
a:= bintr(n-> numtheory[tau](n+1)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 30 2011

a[n_] := Sum[DivisorSigma[0, k+1]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2017 *)

A101509(n) = sum( k=0,n, numdiv(k+1)*binomial(n,k)) \\ M. F. Hasler, Jan 14 2009

a(n) = Sum_{k=0..n, Sum_{i=0..n, if(mod(i+1, k+1)=0, binomial(n, i), 0)}}.
G.f.: 1/x * Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1-x). - Joerg Arndt, Jan 30 2011
a(n) ~ 2^n * (log(n/2) + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 07 2020

A353744 Numbers k such that the k-th composition in standard order has all equal run-lengths.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 45, 48, 49, 50, 52, 54, 58, 63, 64, 65, 66, 68, 69, 70, 72, 76, 77, 80, 81, 82, 88, 89, 96, 97, 98, 101, 102, 104, 105, 108, 109, 127, 128

Offset: 1

Gus Wiseman, Jun 11 2022

nonn

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.

			Composition 2362 in standard order is (3,3,1,1,2,2), with run-lengths (2,2,2), so 2362 is in the sequence.

Standard compositions are listed by A066099.
The version for partitions is A072774, counted by A047966.
These compositions are counted by A329738.
For distinct instead of equal run-lengths we have A351596.
For run-sums instead of lengths we have A353848, counted by A353851.
For distinct run-sums we have A353852, counted by A353850.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353847 represents the composition run-sum transformation.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353833 ranks partitions with all equal run-sums, counted by A304442.
Cf. A029837, A071625, A124767, A175413, A238279, A333381, A333755, A353834, A353849.

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

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