A239312 -id:A239312 - cp's OEIS frontend

A354912 Numbers k such that the k-th composition in standard order is the sequence of run-sums of some other composition.

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 52, 54, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 80, 81, 82, 84, 85, 86, 88, 89, 90, 96, 97, 98, 100, 101, 102, 104, 105, 106, 108

Offset: 0

Gus Wiseman, Jun 22 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.

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 terms and their corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   8: (4)
   9: (3,1)
  10: (2,2)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  21: (2,2,1)
  22: (2,1,2)
For example, the 21st composition in standard order (2,2,1) equals the run-sums of (1,1,2,1), so 21 is in the sequence. On the other hand, no composition has run-sums equal to the 29th composition (1,1,2,1), so 29 is not in the sequence.

The standard compositions used here are A066099, run-sums A353847/A353932.
These are the positions of nonzero terms in A354578.
The complement is A354904, counted by A354909.
These compositions are counted by A354910.
A003242 counts anti-run compositions, ranked by A333489.
A124767 counts runs in standard compositions.
A238279 and A333755 count compositions by number of runs.
A351014 counts distinct runs of standard compositions, firsts A351015.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
Cf. A000120, A029837, A124771, A239312, A333381, A334299, A351597, A353832, A353849, A353860, A354905.

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

A370805 Number of condensed integer partitions of n into parts > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 15, 18, 22, 27, 34, 41, 51, 62, 75, 90, 109, 129, 153, 185, 217, 258, 307, 359, 421, 493, 577, 675, 788, 909, 1062, 1227, 1418, 1633, 1894, 2169, 2497, 2860, 3285, 3754, 4298, 4894, 5587, 6359, 7230, 8215, 9331, 10567, 11965

Offset: 0

Gus Wiseman, Mar 04 2024

nonn

These are partitions without ones such that it is possible to choose a different divisor of each part.

			The a(0) = 1 through a(9) = 6 partitions:
  ()  .  (2)  (3)  (4)    (5)    (6)    (7)      (8)      (9)
                   (2,2)  (3,2)  (3,3)  (4,3)    (4,4)    (5,4)
                                 (4,2)  (5,2)    (5,3)    (6,3)
                                        (3,2,2)  (6,2)    (7,2)
                                                 (3,3,2)  (4,3,2)
                                                 (4,2,2)  (5,2,2)

The version with ones is A239312, complement A370320.
These partitions have as ranks the odd terms of A368110, complement A355740.
The version for prime factors is A370592, complement A370593, post A370807.
The complement without ones is A370804, ranked by the odd terms of A355740.
The version for factorizations is A370814, complement A370813.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A355529, A355739, A367867, A367901, A368110, A368413, A370595, A370806, A370808, A370810.

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1] && Length[Select[Tuples[Divisors/@#],UnsameQ@@#&]]>0&]],{n,0,30}]

More terms from Jinyuan Wang, Feb 14 2025

A370809 Greatest number of multisets that can be obtained by choosing a prime factor of each part of an integer partition of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 4, 4, 6, 4, 6, 6, 6, 6, 8, 6, 8, 8, 9, 8, 10, 9, 12, 10, 12, 12, 12, 12, 16, 13, 16, 16, 18, 16, 20, 18, 20, 20, 24, 20, 24, 24, 24, 26, 30, 26, 30, 30, 32, 32, 36, 32, 36, 36, 40, 38, 42, 40, 45, 44, 48

Offset: 0

Gus Wiseman, Mar 05 2024

nonn

			For the partition (10,6,3,2) there are 4 choices: {2,2,2,3}, {2,2,3,3}, {2,2,3,5}, {2,3,3,5} so a(21) >= 4.
For the partitions of 6 we have the following choices:
  (6): {{2},{3}}
  (51): {}
  (42): {{2,2}}
  (411): {}
  (33): {{3,3}}
  (321): {}
  (3111): {}
  (222): {{2,2,2}}
  (2211): {}
  (21111): {}
  (111111): {}
So a(6) = 2.

For just all divisors (not just prime factors) we have A370808.
The version for factorizations is A370817, for all divisors A370816.
A000041 counts integer partitions, strict A000009.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741, A355744, A355745 choose prime factors of prime indices.
A368413 counts non-choosable factorizations, complement A368414.
A370320 counts non-condensed partitions, ranks A355740.
A370592, A370593, A370594, `A370807 count non-choosable partitions.
Cf. A000792, A048249, A063834, A239312, A319055, A339095, A355529, A355733, A367771, A368100, A370585.

Table[Max[Length[Union[Sort /@ Tuples[If[#==1,{},First/@FactorInteger[#]]& /@ #]]]&/@IntegerPartitions[n]],{n,0,30}]

Terms a(31) onward from Max Alekseyev, Sep 17 2024

A371169 Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 16, 18, 20, 22, 24, 25, 27, 28, 30, 32, 34, 36, 40, 42, 44, 45, 48, 50, 54, 56, 60, 62, 63, 64, 66, 68, 72, 75, 80, 81, 82, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 108, 110, 112, 118, 120, 121, 124, 125, 126, 128, 132, 134, 135

Offset: 1

Gus Wiseman, Mar 16 2024

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 terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    12: {1,1,2}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    22: {1,5}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    32: {1,1,1,1,1}
    34: {1,7}
    36: {1,1,2,2}

The strict version is A370348 counted by A371171.
The case of equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
The complement is A371168, counted by A371173.
The opposite version is A371170.
The version for prime factors instead of divisors on the RHS is A319899.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A003963, A355737, A355739, A355741, A370808, A370813, A370814, A371127.

Select[Range[100],PrimeOmega[#]>=Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A357876 The run-sums of the prime indices of n are not weakly increasing.

Original entry on oeis.org

24, 45, 48, 80, 90, 96, 120, 135, 160, 168, 175, 180, 189, 192, 224, 240, 264, 270, 275, 288, 297, 312, 315, 320, 336, 350, 360, 378, 384, 405, 408, 448, 456, 480, 495, 525, 528, 539, 540, 550, 552, 560, 567, 576, 585, 594, 600, 624, 630, 637, 640, 672, 696

Offset: 1

Gus Wiseman, Oct 17 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:
   24: {1,1,1,2}
   45: {2,2,3}
   48: {1,1,1,1,2}
   80: {1,1,1,1,3}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  120: {1,1,1,2,3}
  135: {2,2,2,3}
  160: {1,1,1,1,1,3}
  168: {1,1,1,2,4}
  175: {3,3,4}
  180: {1,1,2,2,3}
  189: {2,2,2,4}
  192: {1,1,1,1,1,1,2}
For example, the prime indices of 24 are (1,1,1,2), with run-sums (3,2), which are not weakly increasing, so 24 is in the sequence.

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

These are the indices  of rows in A354584 that are not weakly increasing.
The complement is A357875.
These partitions are counted by A357878.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A047966, A118914, A181819, A239312, A275870, A300273, A304442, A325249, A353743-A354912.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!LessEqual@@Total/@Split[primeMS[#]]&]

A371166 Positive integers with fewer divisors (A000005) than distinct divisors of prime indices (A370820).

Original entry on oeis.org

7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 74, 79, 89, 91, 95, 97, 101, 103, 106, 107, 111, 113, 122, 131, 137, 139, 141, 142, 143, 145, 149, 151, 159, 161, 163, 167, 169, 173, 178, 181, 183, 185, 193, 197, 199, 203, 209, 213, 214, 215, 219, 221, 223, 226

Offset: 1

Gus Wiseman, Mar 14 2024

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 terms together with their prime indices begin:
     7: {4}       101: {26}      163: {38}      223: {48}
    13: {6}       103: {27}      167: {39}      226: {1,30}
    19: {8}       106: {1,16}    169: {6,6}     227: {49}
    23: {9}       107: {28}      173: {40}      229: {50}
    29: {10}      111: {2,12}    178: {1,24}    233: {51}
    37: {12}      113: {30}      181: {42}      239: {52}
    43: {14}      122: {1,18}    183: {2,18}    247: {6,8}
    47: {15}      131: {32}      185: {3,12}    251: {54}
    53: {16}      137: {33}      193: {44}      257: {55}
    61: {18}      139: {34}      197: {45}      259: {4,12}
    71: {20}      141: {2,15}    199: {46}      262: {1,32}
    73: {21}      142: {1,20}    203: {4,10}    263: {56}
    74: {1,12}    143: {5,6}     209: {5,8}     265: {3,16}
    79: {22}      145: {3,10}    213: {2,20}    267: {2,24}
    89: {24}      149: {35}      214: {1,28}    269: {57}
    91: {4,6}     151: {36}      215: {3,14}    271: {58}
    95: {3,8}     159: {2,16}    219: {2,21}    281: {60}
    97: {25}      161: {4,9}     221: {6,7}     293: {62}

The RHS is A370820, for prime factors instead of divisors A303975.
For (equal to) instead of (less than) we have A371165, counted by A371172.
For (greater than) instead of (less than) we have A371167.
For prime factors on the LHS we get A371168, counted by A371173.
Other equalities: A319899, A370802 (A371130), A371128, A371177 (A371178).
Other inequalities: A370348 (A371171), A371169, A371170.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts divisor-choosable partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370320 counts non-divisor-choosable partitions, ranks A355740.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. A000792, A003963, A355737, A355739, A355741, A368100, A370808, A371127.

Select[Range[100],Length[Divisors[#]] < Length[Union@@Divisors/@PrimePi/@First/@FactorInteger[#]]&]

A000005(a(n)) < A370820(a(n)).

A371284 Number of integer partitions of n whose distinct parts form the set of divisors of some number.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 5, 8, 9, 11, 12, 16, 18, 23, 25, 32, 36, 42, 47, 57, 62, 73, 81, 96, 106, 123, 132, 154, 168, 190, 207, 240, 259, 293, 317, 359, 388, 434, 469, 529, 574, 635, 688, 768, 826, 915, 987, 1093, 1181, 1302, 1397, 1540, 1662, 1818, 1959, 2149, 2309

Offset: 0

Gus Wiseman, Mar 22 2024

nonn

The Heinz numbers of these partitions are given by A371288.

			The partition y = (10,5,5,5,2,2,1) has distinct parts {1,2,5,10}, which form the set of divisors of 10, so y is counted under a(30).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (11)  (21)   (31)    (221)    (51)      (331)      (71)
             (111)  (211)   (311)    (2211)    (421)      (3311)
                    (1111)  (2111)   (3111)    (511)      (4211)
                            (11111)  (21111)   (2221)     (5111)
                                     (111111)  (22111)    (22211)
                                               (31111)    (221111)
                                               (211111)   (311111)
                                               (1111111)  (2111111)
                                                          (11111111)

The strict case is A054973, ranks A371283 (unsorted version A275700).
These partitions have ranks A371288.
A000005 counts divisors, row-lengths of A027750.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
Cf. A001221, A002865, A239312, A370803, A371172, A371286, A371285.

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

A387118 Number of integer partitions of n without choosable initial intervals.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 6, 8, 13, 19, 28, 37, 52, 70, 97, 130, 172, 224, 293, 378, 492, 630, 806, 1018, 1286, 1609, 2019, 2514, 3131, 3874, 4784, 5872, 7198, 8786, 10712, 13013, 15794, 19100, 23063, 27752, 33341, 39939, 47781, 57013, 67955, 80816, 95992, 113773, 134668

Offset: 0

Gus Wiseman, Sep 02 2025

The initial interval of a nonnegative integer x is the set {1,...,x}.

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

			The partition y = (2,2,1) has initial intervals ({1,2},{1,2},{1}), which are not choosable, so y is counted under a(5).
The a(2) = 1 through a(8) = 13 partitions:
  (11)  (111)  (211)   (221)    (222)     (511)      (611)
               (1111)  (311)    (411)     (2221)     (2222)
                       (2111)   (2211)    (3211)     (3221)
                       (11111)  (3111)    (4111)     (3311)
                                (21111)   (22111)    (4211)
                                (111111)  (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

The complement is counted by A238873, ranks A387112.
The complement for divisors is A239312, ranks A368110.
For divisors instead of initial intervals we have A370320, ranks A355740.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of initial intervals we have A370593, ranks A355529.
These partitions have ranks A387113.
For partitions instead of initial intervals we have A387134.
The complement for partitions is A387328.
For strict partitions instead of initial intervals we have A387137, ranks A387176.
The complement for strict partitions is A387178.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A367902 counts choosable set-systems, complement A367903.
A370582 counts sets with choosable prime factors, complement A370583.
Cf. A005703, A052335, A367867, A367901, A367905, A367907, A370585, A370594, A387111.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[Range/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

More terms from Jinyuan Wang, Sep 05 2025

A387137 Number of integer partitions of n whose parts do not have choosable sets of strict integer partitions.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 9, 14, 20, 29, 39, 56, 74, 101, 134, 178, 232, 305, 392, 508, 646, 825, 1042, 1317, 1649, 2066, 2567, 3190, 3937, 4859, 5960, 7306, 8914, 10863, 13183, 15984, 19304, 23288, 28003, 33631, 40272, 48166, 57453, 68448, 81352, 96568, 114383

Offset: 0

Gus Wiseman, Sep 02 2025

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.
a(n) is the number of integer partitions of n such that it is not possible to choose a sequence of distinct strict integer partitions, one of each part.
Also the number of integer partitions of n with at least one part k whose multiplicity exceeds A000009(k).

			The a(2) = 1 through a(8) = 14 partitions:
  (11)  (111)  (22)    (221)    (222)     (322)      (422)
               (211)   (311)    (411)     (511)      (611)
               (1111)  (2111)   (2211)    (2221)     (2222)
                       (11111)  (3111)    (3211)     (3221)
                                (21111)   (4111)     (3311)
                                (111111)  (22111)    (4211)
                                          (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

The complement for initial intervals is A238873, ranks A387112.
The complement for divisors is A239312, ranks A368110.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
For divisors instead of strict partitions we have A370320, ranks A355740.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of strict partitions we have A370593, ranks A355529.
For initial intervals instead of strict partitions we have A387118, ranks A387113.
For all partitions instead of strict partitions we have A387134, ranks A387577.
These partitions are ranked by A387176.
The complement is counted by A387178, ranks A387177.
The complement for partitions is A387328, ranks A387576.
The version for constant partitions is A387329, ranks A387180.
The complement for constant partitions is A387330, ranks A387181.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A367902 counts choosable set-systems, complement A367903.
Cf. A005703, A052335, A261049, A270995, A276078, A335448, A355535, A367867, A367901, A367905, A383706, A387115.

strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[Select[Tuples[strptns/@#],UnsameQ@@#&]]==0&]],{n,0,15}]

A357878 Number of integer partitions of n whose run-sums are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 40, 55, 79, 104, 150, 196, 270, 350, 467, 600, 786, 997, 1293, 1632, 2077, 2597, 3283, 4067, 5088, 6268, 7769, 9517, 11704, 14238, 17405, 21092, 25598, 30861, 37278, 44729, 53742, 64226, 76811, 91448, 108929, 129174

Offset: 0

Gus Wiseman, Oct 18 2022

nonn

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 a(0) = 0 through a(9) = 8 partitions:
  .  .  .  .  .  (2111)  (21111)  (322)     (3221)     (3222)
                                  (31111)   (32111)    (32211)
                                  (211111)  (311111)   (42111)
                                            (2111111)  (321111)
                                                       (411111)
                                                       (2211111)
                                                       (3111111)
                                                       (21111111)

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

The complement is counted by A304405, ranked by A357875.
Number of rows in A354584 summing to n that are weakly increasing.
The opposite (not weakly increasing) version is A357865, ranked by A357850.
These partitions are ranked by A357876.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A098859, A239312, A275870, A304406, A304428, A304430, A353832, A353864, A353932.

Table[Length[Select[IntegerPartitions[n],!LessEqual@@Total/@Split[Reverse[#]]&]],{n,0,30}]

cp's OEIS Frontend

A354912 Numbers k such that the k-th composition in standard order is the sequence of run-sums of some other composition.

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A370805 Number of condensed integer partitions of n into parts > 1.

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Mathematica

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A370809 Greatest number of multisets that can be obtained by choosing a prime factor of each part of an integer partition of n.

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Mathematica

Extensions

A371169 Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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Mathematica

A357876 The run-sums of the prime indices of n are not weakly increasing.

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Mathematica

A371166 Positive integers with fewer divisors (A000005) than distinct divisors of prime indices (A370820).

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Mathematica

Formula

A371284 Number of integer partitions of n whose distinct parts form the set of divisors of some number.

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Mathematica

A387118 Number of integer partitions of n without choosable initial intervals.

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Mathematica

Extensions

A387137 Number of integer partitions of n whose parts do not have choosable sets of strict integer partitions.

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