A116931 -id:A116931 - cp's OEIS frontend

A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Jan 18 2022

nonn

Differs from A320340 in having 105: (4,3,2), 315: (4,3,2,2), 455: (6,4,3), etc.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with no adjacent prime indices of quotient 1/2.

			The terms and their prime indices begin:
      1: {}            19: {8}             38: {1,8}
      2: {1}           20: {1,1,3}         39: {2,6}
      3: {2}           22: {1,5}           40: {1,1,1,3}
      4: {1,1}         23: {9}             41: {13}
      5: {3}           25: {3,3}           43: {14}
      7: {4}           26: {1,6}           44: {1,1,5}
      8: {1,1,1}       27: {2,2,2}         45: {2,2,3}
      9: {2,2}         28: {1,1,4}         46: {1,9}
     10: {1,3}         29: {10}            47: {15}
     11: {5}           31: {11}            49: {4,4}
     13: {6}           32: {1,1,1,1,1}     50: {1,3,3}
     14: {1,4}         33: {2,5}           51: {2,7}
     15: {2,3}         34: {1,7}           52: {1,1,6}
     16: {1,1,1,1}     35: {3,4}           53: {16}
     17: {7}           37: {12}            55: {3,5}

The version with quotients >= 2 is counted by A000929, sets A018819.
                           <= 2 is A342191, counted by A342094.
                           < 2 is counted by A342096, sets A045690.
                           > 2 is counted by A342098, sets A040039.
The sets version (subsets of prescribed maximum) is counted by A045691.
These partitions are counted by A350837.
The strict case is counted by A350840.
For differences instead of quotients we have A350842, strict A350844.
The complement is A350845, counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000302, A001105, A003000, A018819, A094537, A120641, A154402, A319613, A323093, A337135, A342097, A342095.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[FreeQ[Divide@@@Partition[primeptn[#],2,1],2],{i,2,PrimeOmega[#]}]&]

A325161 Nonprime squarefree numbers not divisible by any two consecutive primes.

Original entry on oeis.org

1, 10, 14, 21, 22, 26, 33, 34, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 95, 106, 110, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 145, 146, 155, 158, 159, 161, 166, 170, 177, 178, 182, 183, 185, 187, 190, 194, 201

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of non-singleton integer partitions into distinct non-consecutive parts (counted by A003114 minus 1).

			The sequence of terms together with their prime indices begins:
   1: {}
  10: {1,3}
  14: {1,4}
  21: {2,4}
  22: {1,5}
  26: {1,6}
  33: {2,5}
  34: {1,7}
  38: {1,8}
  39: {2,6}
  46: {1,9}
  51: {2,7}
  55: {3,5}
  57: {2,8}
  58: {1,10}
  62: {1,11}
  65: {3,6}
  69: {2,9}
  74: {1,12}
  82: {1,13}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001227, A003114, A005117, A025157, A034296, A056239, A073485, A073491, A089995, A112798, A116931, A319630, A325160, A325162.

Select[Range[100],!PrimeQ[#]&&Min@@Differences[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]>1&]

A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 7, 8, 10, 13, 17, 19, 22, 25, 30, 35, 43, 52, 60, 70, 81, 93, 106, 122, 142, 166, 190, 216, 249, 287, 325, 371, 420, 479, 543, 617, 695, 784, 888, 1000, 1126, 1266, 1420, 1594, 1792, 2008, 2247, 2514, 2809, 3135, 3496, 3891, 4332

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(1) = 1 through a(13) = 13 partitions (A..D = 10..13):
  1   2   3   4    5    6    7    8     9     A     B     C     D
              31   32   51   43   53    54    64    65    75    76
                   41        52   62    72    73    74    93    85
                             61   71    81    82    83    A2    94
                                  431   432   91    92    B1    A3
                                        531   532   A1    543   B2
                                              541   641   651   C1
                                                    731   732   643
                                                          741   652
                                                          831   751
                                                                832
                                                                931
                                                                5431

The version for subsets of prescribed maximum is A045691.
The double-free case is A120641.
The non-strict case is A350837, ranked by A350838.
An additive version (differences) is A350844, non-strict A350842.
The non-strict complement is counted by A350846, ranked by A350845.
Versions for prescribed quotients:
  = 2:  A154402, sets A001511.
  != 2: A350840 (this sequence), sets A045691.
  >= 2: A000929, sets A018819.
  <= 2: A342095, non-strict A342094.
  < 2:  A342097, non-strict A342096, sets A045690.
  > 2:  A342098, sets A040039.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A003000, A018819, A303362, A323093, A323094, A337135, A342191.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[#[[i-1]]/#[[i]]!=2,{i,2,Length[#]}]&]],{n,0,30}]

A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 144, 147, 150, 156, 162, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258, 260, 264, 266, 270

Offset: 1

Gus Wiseman, Jan 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at least two adjacent prime indices of quotient 1/2.

			The terms and corresponding partitions begin:
   6: (2,1)
  12: (2,1,1)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  30: (3,2,1)
  36: (2,2,1,1)
  42: (4,2,1)
  48: (2,1,1,1,1)
  54: (2,2,2,1)
  60: (3,2,1,1)
  63: (4,2,2)
  65: (6,3)
  66: (5,2,1)
  72: (2,2,1,1,1)
  78: (6,2,1)
  84: (4,2,1,1)
  90: (3,2,2,1)
  96: (2,1,1,1,1,1)

The complement is A350838, counted by A350837.
The strict complement is counted by A350840.
These partitions are counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A325160 ranks strict partitions with no successions, counted by A003114.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],MemberQ[Divide@@@Partition[primeptn[#],2,1],2]&]

A350846 Number of integer partitions of n with at least two adjacent parts of quotient 2.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 8, 12, 18, 25, 36, 48, 65, 89, 119, 157, 207, 269, 350, 448, 574, 729, 927, 1166, 1465, 1830, 2282, 2827, 3501, 4309, 5300, 6483, 7923, 9641, 11718, 14187, 17155, 20674, 24885, 29860, 35787, 42772, 51054, 60791, 72289, 85772, 101641

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(3) = 1 through a(9) = 12 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (521)      (621)
                       (2211)   (3211)    (3221)     (3321)
                       (21111)  (22111)   (4211)     (4221)
                                (211111)  (22211)    (5211)
                                          (32111)    (22221)
                                          (221111)   (32211)
                                          (2111111)  (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)

The complement is counted by A350837, strict A350840.
The complimentary additive version is A350842, strict A350844.
These partitions are ranked by A350845, complement A350838.
A000041 = integer partitions.
A323092 = double-free integer partitions, ranked by A320340.
Cf. A000929, A003000, A003114, A018819, A045690, A045691, A116931, A120641, A154402, A323093, A342094, A342095, A342096, A342098.

Table[Length[Select[IntegerPartitions[n], MemberQ[Divide@@@Partition[#,2,1],2]&]],{n,0,30}]

A237666 Number of partitions of n that include a pair of consecutive integers.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 7, 9, 15, 20, 32, 40, 61, 78, 112, 142, 199, 250, 341, 428, 568, 710, 930, 1151, 1486, 1835, 2334, 2868, 3615, 4413, 5513, 6706, 8298, 10052, 12359, 14895, 18195, 21857, 26526, 31747, 38337, 45702, 54923, 65272, 78062, 92481, 110168, 130089

Offset: 0

Clark Kimberling, Feb 11 2014

			The qualifying partitions of 8 are 521, 431, 332, 421, 3221, 32111, 22211, 221111, 2111111, so that a(8) = 9.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A116931, A237665.

g:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(g(n-i*j, i-1), j=0..n/i)))
    end:
b:= proc(n, i, l) option remember; `if`(n=0 or i<1, 0,
       b(n, i-1, 0) +add(`if`(i+1=l, g(n-i*j, i-1),
       b(n-i*j, i-1, i)), j=1..n/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 14 2014

Map[Length[Cases[Map[Differences[DeleteDuplicates[#]] &, IntegerPartitions[#]], {_, -1, _}]] &, Range[50]]  (* Peter J. C. Moses, Feb 09 2014 *)
g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Sum[g[n-i*j, i-1], {j, 0, n/i}]]]; b[n_, i_, l_] := b[n, i, l] = If[n==0 || i<1, 0, b[n, i-1, 0] + Sum[If[i+1 == l, g[n-i*j, i-1], b[n-i*j, i-1, i]], {j, 1, n/i}]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 01 2016, after Alois P. Heinz *)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). - Vaclav Kotesovec, Jan 28 2022

Conjecture: for n > 0, a(n) = A000041(n) - A116931(n). - Vaclav Kotesovec, Jan 28 2022

A325162 Squarefree numbers with no two prime indices differing by less than 3.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122, 123, 127, 129, 131

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

A prime index of n is a number m such that prime(m) divides n.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions into distinct parts, no two differing by less than 3 (counted by A025157).

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  22: {1,5}
  23: {9}
  26: {1,6}
  29: {10}
  31: {11}
  33: {2,5}
  34: {1,7}
  37: {12}
  38: {1,8}
  39: {2,6}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001227, A003114, A005117, A025157, A034296, A056239, A073485, A073491, A089995, A112798, A116931, A319630, A325160, A325161.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  if ormap(t -> t[2]>1, F) then return false fi;
  if nops(F) <= 1 then return true fi;
  F:= map(numtheory:-pi,sort(map(t -> t[1],F)));
  min(F[2..-1]-F[1..-2]) >= 3;
end proc:
select(filter, [$1..200]); # Robert Israel, Apr 08 2019

Select[Range[100],Min@@Differences[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]>2&]

A385814 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal proper anti-runs (sequences decreasing by more than 1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 0, 3, 2, 3, 1, 1, 1, 0, 3, 4, 2, 3, 1, 1, 1, 0, 4, 5, 4, 3, 3, 1, 1, 1, 0, 5, 5, 6, 5, 3, 3, 1, 1, 1, 0, 6, 8, 7, 6, 6, 3, 3, 1, 1, 1, 0, 7, 9, 10, 8, 7, 6, 3, 3, 1, 1, 1

Offset: 0

Gus Wiseman, Jul 09 2025

			The partition (8,5,4,2,1) has maximal proper anti-runs ((8,5),(4,2),(1)) so is counted under T(20,3).
The partition (8,5,3,2,2) has maximal proper anti-runs ((8,5,3),(2),(2)) so is also counted under T(20,3).
Row n = 8 counts the following partitions:
  .  8   611  5111  41111  32111   221111  2111111  11111111
     71  521  4211  3221   311111
     62  44   332   2222   22211
     53  431  3311
         422
Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  1  1  1
  0  2  2  1  1  1
  0  3  2  3  1  1  1
  0  3  4  2  3  1  1  1
  0  4  5  4  3  3  1  1  1
  0  5  5  6  5  3  3  1  1  1
  0  6  8  7  6  6  3  3  1  1  1
  0  7  9 10  8  7  6  3  3  1  1  1
  0  9 11 13 12  9  8  6  3  3  1  1  1
  0 10 14 16 15 13 10  8  6  3  3  1  1  1
  0 12 19 18 21 17 14 11  8  6  3  3  1  1  1
  0 14 21 26 23 24 19 15 11  8  6  3  3  1  1  1
  0 17 26 31 33 28 26 20 16 11  8  6  3  3  1  1  1
  0 19 32 37 40 39 31 28 21 16 11  8  6  3  3  1  1  1
  0 23 38 47 50 47 45 34 29 22 16 11  8  6  3  3  1  1  1
  0 26 45 57 61 61 54 48 36 30 22 16 11  8  6  3  3  1  1  1
  0 31 53 71 75 76 70 60 51 37 31 22 16 11  8  6  3  3  1  1  1

Row sums are A000041, strict A000009.
Column k = 1 is A003114.
For anti-runs instead of proper anti-runs we have A268193.
The corresponding rank statistic is A356228.
For proper runs instead of proper anti-runs we have A384881.
For subsets instead of partitions we have A384893, runs A034839.
The strict case is A384905.
For runs instead of proper anti-runs we have A385815.
A007690 counts partitions with no singletons (ranks A001694), complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A047993 counts partitions with max part = length, ranks A106529.
A098859 counts Wilf partitions, complement A336866 (ranks A325992).
A116608 counts partitions by distinct parts.
A116931 counts sparse partitions, ranks A319630.
Cf. A001227, A008284, A089259, A116674, A239455, A325325, A356226, A384880, A384885, A384887, A384906.

Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1>#2+1&]]==k&]],{n,0,10},{k,0,n}]

A385815 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive elements decreasing by 0 or 1.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 5, 2, 0, 0, 0, 0, 7, 4, 0, 0, 0, 0, 0, 8, 7, 0, 0, 0, 0, 0, 0, 10, 12, 0, 0, 0, 0, 0, 0, 0, 13, 16, 1, 0, 0, 0, 0, 0, 0, 0, 15, 25, 2, 0, 0, 0, 0, 0, 0, 0, 0, 18, 34, 4, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jul 09 2025

			The partition (8,5,4,2,1) has maximal runs ((8),(5,4),(2,1)) so is counted under T(20,3).
The partition (8,5,3,2,2) has maximal runs ((8),(5),(3,2,2)) so is also counted under T(20,3).
Row n = 9 counts the following partitions:
  (9)                  (6,3)            (5,3,1)
  (5,4)                (7,2)
  (3,3,3)              (8,1)
  (4,3,2)              (4,4,1)
  (3,2,2,2)            (5,2,2)
  (3,3,2,1)            (6,2,1)
  (2,2,2,2,1)          (7,1,1)
  (3,2,2,1,1)          (4,2,2,1)
  (2,2,2,1,1,1)        (4,3,1,1)
  (3,2,1,1,1,1)        (5,2,1,1)
  (2,2,1,1,1,1,1)      (6,1,1,1)
  (2,1,1,1,1,1,1,1)    (3,3,1,1,1)
  (1,1,1,1,1,1,1,1,1)  (4,2,1,1,1)
                       (5,1,1,1,1)
                       (4,1,1,1,1,1)
                       (3,1,1,1,1,1,1)
Triangle begins:
   1
   0   1
   0   2   0
   0   3   0   0
   0   4   1   0   0
   0   5   2   0   0   0
   0   7   4   0   0   0   0
   0   8   7   0   0   0   0   0
   0  10  12   0   0   0   0   0   0
   0  13  16   1   0   0   0   0   0   0
   0  15  25   2   0   0   0   0   0   0   0
   0  18  34   4   0   0   0   0   0   0   0   0
   0  23  46   8   0   0   0   0   0   0   0   0   0
   0  26  62  13   0   0   0   0   0   0   0   0   0   0
   0  31  82  22   0   0   0   0   0   0   0   0   0   0   0

Row sums are A000041, strict A000009.
Column k = 1 is A034296 (flat or gapless partitions, ranks A066311 or A073491).
For subsets instead of partitions we have A034839, anti-runs A384893.
The strict case appears to be A116674.
For anti-runs instead of runs we have A268193.
The corresponding rank statistic is A287170.
For proper runs instead of runs we have A384881.
For proper anti-runs instead of runs we have A385814.
A007690 counts partitions with no singletons (ranks A001694), complement A183558.
A047993 counts partitions with max part = length, rank A106529.
A098859 counts Wilf partitions, complement A336866 (ranks A325992).
A116608 counts partitions by distinct parts.
A116931 counts sparse partitions, ranks A319630.
Cf. A001227, A008284, A325325, A356226, A384884, A384885, A384887, A384905.

Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1<=#2+1&]]==k&]],{n,0,20},{k,0,n}]

A238422 Number of compositions of n where no consecutive parts differ by 1.

Original entry on oeis.org

1, 1, 2, 2, 5, 7, 15, 23, 43, 70, 128, 214, 383, 651, 1149, 1971, 3457, 5961, 10412, 18011, 31384, 54384, 94639, 164163, 285454, 495452, 861129, 1495126, 2597970, 4511573, 7838280, 13613289, 23649355, 41076088, 71354998, 123939602, 215294730, 373962643, 649597906, 1128352145

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 26 2014

nonn

			The a(6) = 15 such compositions are:
01:  [ 1 1 1 1 1 1 ]
02:  [ 1 1 1 3 ]
03:  [ 1 1 3 1 ]
04:  [ 1 1 4 ]
05:  [ 1 3 1 1 ]
06:  [ 1 4 1 ]
07:  [ 1 5 ]
08:  [ 2 2 2 ]
09:  [ 2 4 ]
10:  [ 3 1 1 1 ]
11:  [ 3 3 ]
12:  [ 4 1 1 ]
13:  [ 4 2 ]
14:  [ 5 1 ]
15:  [ 6 ]

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A116931 (partitions where no consecutive parts differ by 1).

# b(n, i): number of compositions of n where the leftmost part j
#          and i do not have distance 1
b:= proc(n, i) option remember; `if`(n=0, 1,
      add(`if`(abs(i-j)=1, 0, b(n-j, j)), j=1..n))
    end:
a:= n-> b(n, -1):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[Abs[i - j] == 1, 0, b[n - j, j]], {j, 1, n}]]; a[n_] := b[n, -1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 06 2014, after Maple *)

a(n) ~ c * d^n, where c = 0.501153706040308227351395770679776260606990346633815... and d = 1.737029107886986816124470304294547513896522086125645631179... - Vaclav Kotesovec, Feb 26 2014

cp's OEIS Frontend

A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

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A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

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A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

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A350846 Number of integer partitions of n with at least two adjacent parts of quotient 2.

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A325162 Squarefree numbers with no two prime indices differing by less than 3.

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A385814 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal proper anti-runs (sequences decreasing by more than 1).

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A385815 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive elements decreasing by 0 or 1.

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