A356606 -id:A356606 - cp's OEIS frontend

A384889 Number of subsets of {1..n} with all equal lengths of maximal anti-runs (increasing by more than 1).

1, 2, 4, 8, 14, 23, 37, 59, 93, 146, 230, 365, 584, 940, 1517, 2450, 3959, 6404, 10373, 16822, 27298, 44297, 71843, 116429, 188550, 305200, 493930, 799422, 1294108, 2095291, 3392736, 5493168, 8892148, 14390372, 23282110, 37660759, 60914308, 98528312, 159386110

Offset: 0

Gus Wiseman, Jun 18 2025

nonn

			The subset {3,6,7,9,10,12} has maximal anti-runs ((3,6),(7,9),(10,12)), with lengths (2,2,2), so is counted under a(12).
The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {1,3}    {1,2}
                  {2,3}    {1,3}
                  {1,2,3}  {1,4}
                           {2,3}
                           {2,4}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}

Christian Sievers, Table of n, a(n) for n = 0..1000

For runs instead of anti-runs we have A243815, distinct A384175, complement A384176.
For distinct instead or equal lengths we have A384177, ranks A384879.
For partitions instead of subsets we have A384888.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A047966 counts uniform partitions (equal multiplicities), ranks A072774.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A010027, A044813, A047993, A356606, A384880, A384886, A384890.

Table[Length[Select[Subsets[Range[n]],SameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]

lista(n)=Vec(sum(i=1,(n+1)\2,1/(1-x^(2*i-1)/(1-x)^(i-1))-1,1-x+O(x*x^n))/(1-x)^2) \\ Christian Sievers, Jun 20 2025

G.f.: ( Sum_{i>=1} (1/(1-x^(2*i-1)/(1-x)^(i-1))-1) + 1-x ) / (1-x)^2. - Christian Sievers, Jun 21 2025

a(21) and beyond from Christian Sievers, Jun 20 2025

A384887 Number of integer partitions of n with all equal lengths of maximal gapless runs (decreasing by 0 or 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 10, 14, 18, 21, 26, 35, 39, 46, 58, 68, 79, 97, 111, 131, 155, 177, 206, 246, 278, 318, 373, 423, 483, 563, 632, 722, 827, 931, 1058, 1209, 1354, 1528, 1736, 1951, 2188, 2475, 2762, 3097, 3488, 3886, 4342, 4876, 5414, 6038, 6741, 7482

Offset: 0

Gus Wiseman, Jun 15 2025

nonn

			The partition y = (6,5,5,5,3,3,2,1) has maximal gapless runs ((6,5,5,5),(3,3,2,1)), with lengths (4,4), so y is counted under a(30).
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (321)     (2221)     (332)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (3311)
                                               (1111111)  (22211)
                                                          (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

The strict case is A384886, distinct A384178.
For distinct instead of equal lengths we have A384884.
For anti-runs instead of runs we have A384888, distinct A384885.
For subsets instead of strict partitions we have A243815.
Without counting decreases by 0 we get A384904.
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.
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, A325324, A325325, A356226, A356230, A356233, A356234, A384175, A384177, A384880.

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

A384885 Number of integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 8, 9, 13, 15, 18, 22, 28, 31, 38, 45, 53, 62, 74, 86, 105, 123, 146, 171, 208, 242, 290, 340, 399, 469, 552, 639, 747, 862, 999, 1150, 1326, 1514, 1736, 1979, 2256, 2560, 2909, 3283, 3721, 4191, 4726, 5311, 5973, 6691, 7510, 8396, 9395

Offset: 0

Gus Wiseman, Jun 13 2025

nonn

			The partition y = (8,6,3,3,3,1) has maximal anti-runs ((8,6,3),(3),(3,1)), with lengths (3,1,2), so y is counted under a(24).
The partition z = (8,6,5,3,3,1) has maximal anti-runs ((8,6),(5,3),(3,1)), with lengths (2,2,2), so z is not counted under a(26).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)  (3)  (4)    (5)      (6)      (7)      (8)      (9)
                 (3,1)  (4,1)    (4,2)    (5,2)    (5,3)    (6,3)
                        (3,1,1)  (5,1)    (6,1)    (6,2)    (7,2)
                                 (4,1,1)  (3,3,1)  (7,1)    (8,1)
                                          (4,2,1)  (4,2,2)  (4,4,1)
                                          (5,1,1)  (4,3,1)  (5,2,2)
                                                   (5,2,1)  (5,3,1)
                                                   (6,1,1)  (6,2,1)
                                                            (7,1,1)

For subsets instead of strict partitions we have A384177, for runs A384175.
The strict case is A384880.
For runs instead of anti-runs we have A384884, strict A384178.
For equal instead of distinct lengths we have A384888, for runs 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, A047966, A242882, A287170, A325324, A325325, A329739, A356226, A356230, A356234, A384886.

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

A384891 Number of permutations of {1..n} with all distinct lengths of maximal runs (increasing by 1).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 23, 25, 43, 63, 345, 365, 665, 949, 1513, 8175, 9003, 15929, 23399, 36949, 51043, 293715, 314697, 570353, 826817, 1318201, 1810393, 2766099, 14180139, 15600413, 27707879, 40501321, 63981955, 88599903, 134362569, 181491125, 923029217

Offset: 0

Gus Wiseman, Jun 19 2025

nonn

			The permutation (1,2,6,7,8,9,3,4,5) has maximal runs ((1,2),(6,7,8,9),(3,4,5)), with lengths (2,4,3), so is counted under a(9).
The a(0) = 1 through a(7) = 25 permutations:
  ()  (1)  (12)  (123)  (1234)  (12345)  (123456)  (1234567)
                 (231)  (2341)  (23451)  (123564)  (1234675)
                 (312)  (4123)  (34512)  (123645)  (1234756)
                                (45123)  (124563)  (1245673)
                                (51234)  (126345)  (1273456)
                                         (145623)  (1456723)
                                         (156234)  (1672345)
                                         (231456)  (2314567)
                                         (234156)  (2345167)
                                         (234561)  (2345671)
                                         (312456)  (3124567)
                                         (345126)  (3456127)
                                         (345612)  (3456712)
                                         (412356)  (4567123)
                                         (451236)  (4567231)
                                         (456231)  (4567312)
                                         (456312)  (5123467)
                                         (561234)  (5612347)
                                         (562341)  (5671234)
                                         (564123)  (6712345)
                                         (612345)  (6723451)
                                         (634512)  (6751234)
                                         (645123)  (7123456)
                                                   (7345612)
                                                   (7561234)

Christian Sievers, Table of n, a(n) for n = 0..5000

Counting by number of maximal anti-runs gives A010027, for runs A123513.
For subsets instead of permutations we have A384175, complement A384176.
For partitions we have A384884 (anti-runs A384885), strict A384178 (anti-runs A384880).
For equal instead of distinct lengths we have A384892.
For anti-runs instead of runs we have A384907.
A000041 counts integer partitions, strict A000009.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A356606 counts strict partitions without a neighborless part, complement A356607.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000255, A044813, A072574, A242882, A287170, A325324, A325325, A328592, A329739, A351202, A384177, A384886.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

lista(n)=my(b(n)=sum(i=0,n-1,(-1)^i*(n-i)!*binomial(n-1,i)), d=floor(sqrt(2*n)), p=prod(i=1,n,1+x*y^i,1+O(y*y^n)*((1-x^(n+1))/(1-x))+O(x*x^d))); Vec(1+sum(i=1,d,i!*b(i)*polcoef(p,i))) \\ Christian Sievers, Jun 22 2025

a(n) = Sum_{k=1..n} ( T(n,k) * A000255(k-1) ) for n>=1, where T(n,k) is the number of compositions of n into k distinct parts (cf. A072574). - Christian Sievers, Jun 22 2025

a(11) and beyond from Christian Sievers, Jun 22 2025

A356734 Heinz numbers of integer partitions with at least one neighborless part.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 78, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Aug 26 2022

nonn

First differs from A319630 in lacking 1 and having 42 (prime indices: {1,2,4}).
A part x is neighborless if neither x - 1 nor x + 1 are parts.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}

These partitions are counted by A356236.
The singleton case is A356237, counted by A356235 (complement A355393).
The strict case is counted by A356607, complement A356606.
The complement is A356736, counted by A355394.
A001221 counts distinct prime factors, 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).
Cf. A000005, A286470, A287170 (firsts A066205), 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[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A384888 Number of integer partitions of n with all equal lengths of maximal anti-runs (decreasing by more than 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 10, 13, 17, 20, 24, 32, 36, 44, 55, 64, 75, 92, 105, 125, 147, 169, 195, 231, 263, 303, 351, 401, 458, 532, 600, 686, 784, 889, 1010, 1152, 1296, 1468, 1662, 1875, 2108, 2384, 2669, 3001, 3373, 3775, 4222, 4734, 5278, 5896, 6576, 7322

Offset: 0

Gus Wiseman, Jun 15 2025

nonn

			The partition y = (10,6,6,4,3,1) has maximal anti-runs ((10,6),(6,4),(3,1)), with lengths (2,2,2), so y is counted under a(30).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (321)     (2221)     (332)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

The strict case is new, distinct A384880.
For distinct instead of equal lengths we have A384885.
For runs instead of anti-runs we have A384887, distinct A384884.
For subsets instead of strict partitions we have A384889, distinct A384177, runs A243815.
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, A325325, A356226, A384175, A384176, A384178, A384886.

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

A356736 Heinz numbers of integer partitions with no neighborless parts.

Original entry on oeis.org

1, 6, 12, 15, 18, 24, 30, 35, 36, 45, 48, 54, 60, 72, 75, 77, 90, 96, 105, 108, 120, 135, 143, 144, 150, 162, 175, 180, 192, 210, 216, 221, 225, 240, 245, 270, 288, 300, 315, 323, 324, 360, 375, 384, 385, 405, 420, 432, 437, 450, 462, 480, 486, 525, 539, 540

Offset: 1

Gus Wiseman, Aug 31 2022

nonn

First differs from A066312 in having 1 and lacking 462.
First differs from A104210 in having 1 and lacking 42.
A part x is neighborless iff neither x - 1 nor x + 1 are parts.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  30: {1,2,3}
  35: {3,4}
  36: {1,1,2,2}
  45: {2,2,3}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  60: {1,1,2,3}
  72: {1,1,1,2,2}
  75: {2,3,3}
  77: {4,5}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

These partitions are counted by A355394.
The singleton case is the complement of A356237.
The singleton case is counted by A355393, complement A356235.
The strict complement is A356606, counted by A356607.
The complement is A356734, counted by A356236.
A000041 counts integer partitions, strict A000009.
A001221 counts distinct prime factors, 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.
Cf. A066312, A286470, A287170 (firsts A066205), A328171, A328187, A328221, A328335, A356231, A356234.

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

A385213 Number of maximal runs of consecutive parts increasing by 1 in the prime indices of n (with multiplicity).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 22 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 24 are {1,1,1,2}, with maximal runs ((1),(1),(1,2)), so a(24) = 3.

Positions of first appearances are A000079.
For binary instead of prime indices we have A069010 (for anti-runs A384890).
For anti-runs instead of runs we have A384906.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A002110, A048767, A130091, A300820, A356606, A351202, A382525, A384893.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Split[prix[n],#2==#1+1&]],{n,100}]

A384907 Number of permutations of {1..n} with all distinct lengths of maximal anti-runs (not increasing by 1).

Original entry on oeis.org

1, 1, 1, 5, 17, 97, 587, 4291, 33109, 319967, 3106433, 35554459, 419889707, 5632467097, 77342295637, 1201240551077, 18804238105133, 328322081898745, 5832312989183807, 113154541564902427, 2229027473451951265, 47899977701182298255, 1037672943682453127645

Offset: 0

Gus Wiseman, Jun 21 2025

nonn

			The permutation (1,2,4,3,5,7,8,6,9) has maximal anti-runs ((1),(2,4,3,5,7),(8,6,9)), with lengths (1,5,3), so is counted under a(9).
The a(0) = 1 through a(4) = 17 permutations:
  ()  (1)  (2,1)  (1,3,2)  (1,2,4,3)
                  (2,1,3)  (1,3,2,4)
                  (2,3,1)  (1,4,2,3)
                  (3,1,2)  (1,4,3,2)
                  (3,2,1)  (2,1,3,4)
                           (2,1,4,3)
                           (2,3,1,4)
                           (2,4,1,3)
                           (2,4,3,1)
                           (3,1,4,2)
                           (3,2,1,4)
                           (3,2,4,1)
                           (3,4,2,1)
                           (4,1,3,2)
                           (4,2,1,3)
                           (4,3,1,2)
                           (4,3,2,1)

Christian Sievers, Table of n, a(n) for n = 0..449

For subsets instead of permutations we have A384177.
For strict partitions we have A384880, for runs A384178.
For partitions we have A384885, for runs A384884.
For runs instead of anti-runs we have A384891.
A010027 counts permutations by maximal anti-runs, for runs A123513.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000255, A044813, A072574, A242882, A325325, A328592, A329739, A351202, A356606, A384886, A384892.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]

a(n)=if(n,my(b(n)=sum(i=0,n-1,(-1)^i*(n-i)!*binomial(n-1,i)), d=floor(sqrt(2*n)), p=polcoef(prod(i=1,n,1+x*y^i,1+O(y*y^n)*((1-x^(d+1))/(1-x))),n,y)); sum(i=1,d,b(n+1-i)*i!*polcoef(p,i)),1) \\ Christian Sievers, Jun 22 2025

a(n) = Sum_{k=1..n} ( T(n,k) * A000255(n-k) ) for n>=1, where T(n,k) is the number of compositions of n into k distinct parts (cf. A072574).

a(11) and beyond from Christian Sievers, Jun 22 2025

A356733 Number of neighborless parts in the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 2, 2, 1, 0, 1, 2, 1, 2, 1, 0, 1, 1, 2, 2, 0, 0, 1, 2, 2, 2, 1, 1, 1, 2, 0, 2, 1, 0, 1, 2, 2, 2, 1, 0, 2, 2, 2, 2, 1, 0, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 0, 1, 2, 0, 2, 0, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 0, 2, 2, 2, 2, 2, 0, 1, 2, 2, 2, 1, 1, 1, 2, 0

Offset: 1

Gus Wiseman, Aug 26 2022

nonn

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

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The prime indices of 42 are {1,2,4}, of which only 4 is neighborless, so a(42) = 1.
The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 0.
The prime indices of 1300 are {1,1,3,3,6}, with neighborless parts {1,3,6}, so a(1300) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of first appearances are 1 followed by A066205.
Dominated by A287170 (firsts also A066205).
Positions of terms > 0 are A356734.
The complement is counted by A356735.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together prime indices.
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.
A355393 counts partitions w/o a neighborless singleton, complement A356235.
A355394 counts partitions w/o a neighborless part, complement A356236.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356607 counts strict partitions w/ a neighborless part, complement A356606.
Cf. A000005, A066312, A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234, A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Union[primeMS[n]],!MemberQ[primeMS[n],#-1]&&!MemberQ[primeMS[n],#+1]&]],{n,100}]

A356733(n) = if(1==n,0,my(pis=apply(primepi,factor(n)[,1])); sum(i=1, #pis, ((n%prime(pis[i]+1)) && (pis[i]==1 || (n%prime(pis[i]-1)))))); \\ Antti Karttunen, Jan 28 2025

a(n) = A001221(n) - A356735(n).

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

cp's OEIS Frontend

A384889 Number of subsets of {1..n} with all equal lengths of maximal anti-runs (increasing by more than 1).

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A384887 Number of integer partitions of n with all equal lengths of maximal gapless runs (decreasing by 0 or 1).

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A384885 Number of integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

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A384891 Number of permutations of {1..n} with all distinct lengths of maximal runs (increasing by 1).

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A356734 Heinz numbers of integer partitions with at least one neighborless part.

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A384888 Number of integer partitions of n with all equal lengths of maximal anti-runs (decreasing by more than 1).

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A356736 Heinz numbers of integer partitions with no neighborless parts.

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A385213 Number of maximal runs of consecutive parts increasing by 1 in the prime indices of n (with multiplicity).

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A384907 Number of permutations of {1..n} with all distinct lengths of maximal anti-runs (not increasing by 1).

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