A356606 -id:A356606 - cp's OEIS frontend

A384877 Irregular triangle read by rows where row k lists the lengths of maximal anti-runs (increasing by more than 1) in the binary indices of n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 1, 1, 2, 2, 3, 3, 1, 3, 1, 2, 2, 2

Offset: 0

Gus Wiseman, Jun 17 2025

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of 182 are {2,3,5,6,8}, with maximal anti-runs ((2),(3,5),(6,8)) so row 182 is (1,2,2).
Triangle begins:
   0: ()
   1: (1)
   2: (1)
   3: (1,1)
   4: (1)
   5: (2)
   6: (1,1)
   7: (1,1,1)
   8: (1)
   9: (2)
  10: (2)
  11: (1,2)
  12: (1,1)
  13: (2,1)
  14: (1,1,1)
  15: (1,1,1,1)

Row-sums are A000120.
Positions of rows of the form (1,1,...) are A023758.
Positions of first appearances of each distinct row appear to be A052499.
For runs instead of anti-runs we have A245563, reverse A245562.
Row-lengths are A384890.
A355394 counts partitions without a neighborless part, singleton case A355393.
A356606 counts strict partitions without a neighborless part, complement A356607.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
Cf. A044813, A048793, A069010, A164707, A243815, A246029, A328592, A384177, A384877, A384879, A384893.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length/@Split[bpe[n],#2!=#1+1&],{n,0,100}]

A384890 Number of maximal anti-runs (increasing by more than 1) in the binary indices of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 17 2025

nonn

First differs from A272604 at a(51) = 3, A272604(51) = 2.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Do all constant runs in this sequence have lengths 1, 2, or 3?

			The binary indices of 51 are {1,2,5,6}, with maximal anti-runs ((1),(2,5),(6)), so a(51) = 3.

For runs instead of anti-runs we have A069010 = run-lengths of A245563 (reverse A245562).
Row-lengths of A384877, firsts A384878.
For prime indices instead of binary indices we have A384906.
A000120 counts binary indices.
A356606 counts strict partitions without a neighborless part, complement A356607.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
Cf. A044813, A048793, A052499, A164707, A243815, A300820, A328592, A384177, A384879, A384893.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Split[bpe[n],#2!=#1+1&]],{n,0,100}]

A384884 Number of integer partitions of n with all distinct lengths of maximal gapless runs (decreasing by 0 or 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 35, 46, 60, 79, 104, 131, 170, 215, 271, 342, 431, 535, 670, 830, 1019, 1258, 1547, 1881, 2298, 2787, 3359, 4061, 4890, 5849, 7010, 8361, 9942, 11825, 14021, 16558, 19561, 23057, 27084, 31821, 37312, 43627, 50999, 59500, 69267

Offset: 0

Gus Wiseman, Jun 13 2025

nonn

			The partition y = (6,6,4,3,3,2) has maximal gapless runs ((6,6),(4,3,3,2)), with lengths (2,4), so y is counted under a(24).
The a(1) = 1 through a(8) = 18 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (222)     (322)      (332)
                    (1111)  (311)    (321)     (331)      (422)
                            (2111)   (411)     (421)      (431)
                            (11111)  (2211)    (511)      (521)
                                     (3111)    (2221)     (611)
                                     (21111)   (3211)     (2222)
                                     (111111)  (4111)     (3221)
                                               (22111)    (4211)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For subsets instead of strict partitions we have A384175.
The strict case is A384178, for anti-runs A384880.
For anti-runs we have A384885.
For equal instead of distinct lengths we have A384887.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A355394 counts partitions without a neighborless part, singleton case A355393.
A356236 counts partitions with a neighborless part, singleton case A356235.
A356606 counts strict partitions without a neighborless part, complement A356607.
Cf. A008284, A044813, A047993, A242882, A287170, A325324, A325325, A356226, A356230, A356233, A356234, A384176, A384177, A384886.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2>=#1-1&]&]],{n,0,15}]

A356607 Number of strict integer partitions of n with at least one neighborless part.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 13, 17, 20, 24, 30, 36, 41, 52, 60, 71, 84, 100, 114, 137, 158, 183, 214, 248, 283, 330, 379, 432, 499, 570, 648, 742, 846, 955, 1092, 1234, 1395, 1580, 1786, 2005, 2270, 2548, 2861, 3216, 3610, 4032, 4526, 5055, 5642, 6304, 7031, 7820, 8720, 9694

Offset: 0

Gus Wiseman, Aug 26 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts.

			The a(0) = 0 through a(9) = 6 partitions:
  .  (1)  (2)  (3)  (4)   (5)   (6)   (7)    (8)    (9)
                    (31)  (41)  (42)  (52)   (53)   (63)
                                (51)  (61)   (62)   (72)
                                      (421)  (71)   (81)
                                             (431)  (531)
                                             (521)  (621)

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, A356607.py

This is the strict case of A356235 and A356236.
The complement is counted by A356606, non-strict A355393 and A355394.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A073492, A137921, A325160, A328171, A328172, A328187, A328220, A328221.

Table[Length[Select[IntegerPartitions[n],Function[ptn,UnsameQ@@ptn&&Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

a(31)-a(59) from Lucas A. Brown, Sep 09 2022

A355394 Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 3, 6, 6, 10, 11, 16, 18, 25, 30, 38, 47, 59, 74, 90, 112, 136, 171, 203, 253, 299, 372, 438, 536, 631, 767, 900, 1085, 1271, 1521, 1774, 2112, 2463, 2910, 3389, 3977, 4627, 5408, 6276, 7304, 8459, 9808, 11338, 13099, 15112, 17404, 20044, 23018, 26450, 30299, 34746, 39711, 45452, 51832

Offset: 0

Gus Wiseman, Aug 26 2022

nonn

These are partitions without a neighborless part, where a part x is neighborless if neither x - 1 nor x + 1 are parts. The first counted partition that does not cover an interval is (5,4,2,1).

			The a(0) = 1 through a(9) = 11 partitions:
  ()  .  .  (21)  (211)  (32)    (321)    (43)      (332)      (54)
                         (221)   (2211)   (322)     (3221)     (432)
                         (2111)  (21111)  (2221)    (22211)    (3222)
                                          (3211)    (32111)    (3321)
                                          (22111)   (221111)   (22221)
                                          (211111)  (2111111)  (32211)
                                                               (222111)
                                                               (321111)
                                                               (2211111)
                                                               (21111111)

Lucas A. Brown, Table of n, a(n) for n = 0..100
Lucas A. Brown, A355394.py

The singleton case is A355393, complement A356235.
The complement is counted by A356236, ranked by A356734.
The strict case is A356606, complement A356607.
These partitions are ranked by A356736.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066312, A073491, A077855, A328171, A328172, A328187, A328221, A356233, A356237.

Table[Length[Select[IntegerPartitions[n],Function[ptn,!Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

a(n) = A000041(n) - A356236(n).

a(31)-a(59) from Lucas A. Brown, Sep 04 2022

A356237 Heinz numbers of integer partitions with a neighborless singleton.

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93

Offset: 1

Gus Wiseman, Aug 24 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.
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.
Also numbers that, for some prime index x, are not divisible by prime(x)^2, prime(x - 1), or prime(x + 1). Here, a prime index of n is a number m such that prime(m) divides n.

			The terms together with their prime indices begin:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  26: {1,6}
  28: {1,1,4}

The complement is counted by A355393.
These partitions are counted by A356235.
Not requiring a singleton gives A356734.
A001221 counts distinct prime factors, with sum A001414.
A003963 multiplies together the prime indices of n.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356236 counts partitions with a neighborless part, complement A355394.
A356607 counts strict partitions w/ a neighborless part, complement A356606.
Cf. A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Function[ptn,Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A384879 Numbers whose binary indices have all distinct lengths of maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 11, 13, 16, 17, 18, 19, 20, 21, 22, 25, 26, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 49, 50, 52, 53, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 88, 97, 98, 100, 101, 104, 105, 106, 128, 129, 130

Offset: 1

Gus Wiseman, Jun 17 2025

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of 813 are {1,3,4,6,9,10}, with maximal anti-runs ((1,3),(4,6,9),(10)), with lengths (2,3,1), so 813 is in the sequence.
The terms together with their binary expansions and binary indices begin:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    5:     101 ~ {1,3}
    8:    1000 ~ {4}
    9:    1001 ~ {1,4}
   10:    1010 ~ {2,4}
   11:    1011 ~ {1,2,4}
   13:    1101 ~ {1,3,4}
   16:   10000 ~ {5}
   17:   10001 ~ {1,5}
   18:   10010 ~ {2,5}
   19:   10011 ~ {1,2,5}
   20:   10100 ~ {3,5}
   21:   10101 ~ {1,3,5}
   22:   10110 ~ {2,3,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}

Subsets of this type are counted by A384177, for runs A384175 (complement A384176).
These are the indices of strict rows in A384877, see A384878, A245563, A245562, A246029.
A000120 counts binary indices.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A356606 counts strict partitions without a neighborless part, complement A356607.
A384890 counts maximal anti-runs in binary indices, runs A069010.
Cf. A023758, A044813, A048793, A164707, A242882, A243815, A325325, A328592, A384879, A384884, A384886, A384893.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],UnsameQ@@Length/@Split[bpe[#],#2!=#1+1&]&]

A355393 Number of integer partitions of n such that, for all parts x of multiplicity 1, either x - 1 or x + 1 is also a part.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 6, 7, 10, 14, 17, 23, 32, 39, 51, 67, 83, 105, 134, 165, 206, 256, 312, 385, 475, 573, 697, 849, 1021, 1231, 1483, 1771, 2121, 2534, 3007, 3575, 4245, 5008, 5914, 6979, 8198, 9626, 11292, 13201, 15430, 18010, 20960, 24389, 28346, 32855, 38066

Offset: 0

Gus Wiseman, Aug 26 2022

nonn

These are partitions without a neighborless singleton, where a part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.

			The a(0) = 1 through a(8) = 10 partitions:
  ()  .  (11)  (21)   (22)    (32)     (33)      (43)       (44)
               (111)  (211)   (221)    (222)     (322)      (332)
                      (1111)  (2111)   (321)     (2221)     (2222)
                              (11111)  (2211)    (3211)     (3221)
                                       (21111)   (22111)    (3311)
                                       (111111)  (211111)   (22211)
                                                 (1111111)  (32111)
                                                            (221111)
                                                            (2111111)
                                                            (11111111)

This is the singleton case of A355394, complement A356236.
The complement is counted by A356235.
These partitions are ranked by the complement of A356237.
The strict case is A356606, complement A356607.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A073491, A325160, A328171, A328172, A328187, A328220, A356233.

Table[Length[Select[IntegerPartitions[n],Function[ptn,!Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

A356235 Number of integer partitions of n with a neighborless singleton.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 5, 8, 12, 16, 25, 33, 45, 62, 84, 109, 148, 192, 251, 325, 421, 536, 690, 870, 1100, 1385, 1739, 2161, 2697, 3334, 4121, 5071, 6228, 7609, 9303, 11308, 13732, 16629, 20101, 24206, 29140, 34957, 41882, 50060, 59745, 71124, 84598, 100365

Offset: 0

Gus Wiseman, Aug 23 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once. Examples of partitions with a neighborless singleton are: (3), (3,1), (3,1,1), (3,3,1). Examples of partitions without a neighborless singleton are: (3,3,1,1), (4,3,1,1), (3,2,1), (2,1), (3,3).

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)  (3)  (4)   (5)    (6)     (7)      (8)
                 (31)  (41)   (42)    (52)     (53)
                       (311)  (51)    (61)     (62)
                              (411)   (331)    (71)
                              (3111)  (421)    (422)
                                      (511)    (431)
                                      (4111)   (521)
                                      (31111)  (611)
                                               (4211)
                                               (5111)
                                               (41111)
                                               (311111)

The complement is counted by A355393.
This is the singleton case of A356236, complement A355394.
These partitions are ranked by A356237.
The strict case is A356607, complement A356606.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066205, A325160, A328171, A328172, A328187, A328221, A356233.

Table[Length[Select[IntegerPartitions[n],Min@@Length/@Split[Reverse[#],#1>=#2-1&]==1&]],{n,0,30}]

A356236 Number of integer partitions of n with a neighborless part.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 8, 9, 16, 20, 31, 40, 59, 76, 105, 138, 184, 238, 311, 400, 515, 656, 831, 1052, 1322, 1659, 2064, 2572, 3182, 3934, 4837, 5942, 7264, 8872, 10789, 13109, 15865, 19174, 23105, 27796, 33361, 39956, 47766, 56985, 67871, 80675, 95750, 113416

Offset: 0

Gus Wiseman, Aug 24 2022

nonn

A part x of a partition is neighborless if neither x - 1 nor x + 1 are parts.

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (41)     (33)      (52)
                    (31)    (311)    (42)      (61)
                    (1111)  (11111)  (51)      (331)
                                     (222)     (421)
                                     (411)     (511)
                                     (3111)    (4111)
                                     (111111)  (31111)
                                               (1111111)

The complement is counted by A355394, singleton case A355393.
The singleton case is A356235, ranked by A356237.
The strict case is A356607, complement A356606.
These partitions are ranked by the complement of A356736.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066205, A112798, A319630, A325160, A328171, A328172, A328187, A328221.

Table[Length[Select[IntegerPartitions[n],Function[ptn,Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

a(n) = A000041(n) - A355394(n).

cp's OEIS Frontend

A384877 Irregular triangle read by rows where row k lists the lengths of maximal anti-runs (increasing by more than 1) in the binary indices of n.

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A384890 Number of maximal anti-runs (increasing by more than 1) in the binary indices of n.

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A384884 Number of integer partitions of n with all distinct lengths of maximal gapless runs (decreasing by 0 or 1).

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A356607 Number of strict integer partitions of n with at least one neighborless part.

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A355394 Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.

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A356237 Heinz numbers of integer partitions with a neighborless singleton.

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A384879 Numbers whose binary indices have all distinct lengths of maximal anti-runs (increasing by more than 1).

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A355393 Number of integer partitions of n such that, for all parts x of multiplicity 1, either x - 1 or x + 1 is also a part.

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A356235 Number of integer partitions of n with a neighborless singleton.

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