A200745 -id:A200745 - cp's OEIS frontend

A300702 Number of compositions (ordered partitions) of n into parts that do not divide n.

1, 0, 0, 0, 0, 2, 0, 7, 2, 7, 7, 54, 2, 143, 33, 47, 30, 986, 23, 2583, 58, 1018, 828, 17710, 32, 23866, 3917, 14586, 1368, 317810, 248, 832039, 5902, 188953, 85038, 1505979, 602, 14930351, 393663, 2350986, 13524, 102334154, 16401, 267914295, 431711, 4438212, 8400611, 1836311902

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

			a(7) = 7 because we have [5, 2], [4, 3], [3, 4], [3, 2, 2], [2, 5], [2, 3, 2] and [2, 2, 3].

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Index entries for sequences related to compositions

Cf. A011782, A098743, A100346, A200745.

a:= proc(m) option remember; local b; b:= proc(n)
      option remember; `if`(n=0, 1, add(`if`(
      irem(m, j)=0, 0, b(n-j)), j=2..n)) end; b(m)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 11 2018

Table[SeriesCoefficient[1/(1 - Sum[Boole[Mod[n, k] != 0] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 47}]

A294266 Number of partitions of n into distinct squares that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 1, 3, 0, 0, 2, 2, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 1, 0, 0, 2, 3, 0, 0, 1, 1, 0, 1, 2, 1, 1, 0, 2, 2, 0, 0, 3, 4

Offset: 0

Ilya Gutkovskiy, Oct 26 2017

			a(29) = 2 because we have [25, 4] and [16, 9, 4].

Cf. A033461, A200745, A284345, A294265, A371968.

f:= proc(n) local P,k,x;
P:= 1;
for k from 2 to floor(sqrt(n)) do
  if n mod k^2 = 0 then next fi;
  P:= series(P*(1+x^(k^2)),x,n+1);
od;
coeff(P,x,n)
end proc:
map(f, [$0..200]); # Robert Israel, Apr 15 2024

Table[SeriesCoefficient[Product[1 + Boole[Mod[n, k] > 0 && OddQ[DivisorSigma[0, k]]] x^k, {k, 1, n}], {x, 0, n}], {n, 0, 110}]

A300584 Number of partitions of n into distinct prime power parts (not including 1) that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 2, 1, 4, 1, 5, 2, 4, 3, 8, 2, 11, 4, 6, 6, 18, 3, 13, 10, 10, 8, 35, 7, 42, 9, 20, 24, 22, 10, 72, 33, 35, 11, 102, 21, 120, 29, 26, 65, 161, 16, 117, 56, 93, 52, 246, 43, 113, 36, 139, 160, 370, 39, 422, 208, 102, 62, 216, 110, 613, 145, 305, 130, 780, 57, 878

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

			a(9) = 2 because we have [7, 2] and [5, 4].

Index entries for sequences related to partitions

Cf. A054685, A200745, A209402, A246655, A300580.

Table[SeriesCoefficient[Product[(1 + Boole[Mod[n, k] != 0 && PrimePowerQ[k]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 73}]

A339660 Number of strict integer partitions of n with no 1's and a part divisible by all the other parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 3, 5, 2, 5, 5, 4, 5, 7, 3, 5, 6, 5, 5, 9, 4, 7, 6, 6, 9, 11, 6, 9, 10, 9, 10, 12, 6, 11, 12, 10, 12, 16, 9, 15, 16, 12, 14, 18, 14, 16, 18, 14, 15, 22, 11, 16, 20, 13, 21, 23, 15, 21, 24, 21, 21, 31, 14, 24

Offset: 0

Gus Wiseman, Apr 19 2021

nonn

Alternative name: Number of strict integer partitions of n with no 1's that are empty or have greatest part divisible by all the other parts.

			The a(n) partitions for n = 14, 12, 18, 24, 30, 39, 36:
  (14)     (12)    (18)      (24)        (30)        (39)          (36)
  (12,2)   (8,4)   (12,6)    (16,8)      (24,6)      (36,3)        (27,9)
  (8,4,2)  (9,3)   (15,3)    (18,6)      (25,5)      (26,13)       (30,6)
           (10,2)  (16,2)    (20,4)      (27,3)      (27,9,3)      (32,4)
                   (12,4,2)  (21,3)      (28,2)      (28,7,4)      (33,3)
                             (22,2)      (20,10)     (30,6,3)      (34,2)
                             (12,6,4,2)  (18,9,3)    (24,12,3)     (24,12)
                                         (24,4,2)    (24,8,4,3)    (24,8,4)
                                         (16,8,4,2)  (20,10,5,4)   (18,9,6,3)
                                                     (24,6,4,3,2)  (24,6,4,2)
                                                                   (20,10,4,2)

The dual version is A098965 (non-strict: A083711).
The non-strict version is A339619 (Heinz numbers: complement of A343337).
The version with 1's allowed is A343347 (non-strict: A130689).
The case without a part dividing all the other parts is A343380.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
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. A083710, A097986, A098743, A200745, A264401, A339563, A341450, A343337, A343341, A343344, A343378.

Table[If[n==0,1,Length[Select[IntegerPartitions[n],FreeQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]]],{n,0,30}]

A300586 Number of partitions of n into distinct squarefree parts that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 4, 2, 2, 4, 6, 2, 8, 4, 6, 6, 15, 4, 11, 10, 12, 8, 30, 3, 38, 24, 17, 24, 23, 14, 70, 36, 37, 23, 102, 8, 122, 49, 39, 80, 177, 38, 136, 54, 113, 101, 297, 60, 152, 102, 192, 226, 485, 28, 571, 312, 200, 390, 338, 84, 908, 393, 507, 104, 1229, 241, 1421

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

			a(14) = 2 because we have [11, 3] and [6, 5, 3].

Index entries for sequences related to partitions

Cf. A005117, A087188, A200745, A209402, A225245, A300584, A300585.

Table[SeriesCoefficient[Product[(1 + Boole[Mod[n, k] != 0 && SquareFreeQ[k]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 73}]

A343348 Irregular triangle read by rows where T(n,k) is the number of strict integer partitions of n with least gap k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 1, 3, 1, 1, 0, 3, 2, 1, 0, 5, 2, 1, 0, 5, 3, 1, 0, 1, 7, 3, 1, 1, 0, 8, 4, 2, 1, 0, 10, 5, 2, 1, 0, 12, 6, 3, 1, 0, 15, 7, 3, 1, 0, 1, 17, 9, 4, 1, 1, 0, 21, 10, 4, 2, 1, 0, 25, 12, 6, 2, 1, 0, 29, 15, 6, 3, 1, 0, 35, 17, 8, 3, 1, 0

Offset: 0

Gus Wiseman, Apr 18 2021

The least gap (or mex) of a partition is the least positive integer that is not a part.

Row lengths are chosen to be consistent with the non-strict case A264401.

			Triangle begins:
   1
   0   1
   1   0
   1   0   1
   1   1   0
   2   1   0
   2   1   0   1
   3   1   1   0
   3   2   1   0
   5   2   1   0
   5   3   1   0   1
   7   3   1   1   0
   8   4   2   1   0
  10   5   2   1   0
  12   6   3   1   0
  15   7   3   1   0   1

George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.

Row sums are A000009.
Row lengths are A002024.
Column k = 1 is A025147.
Column k = 2 is A025148.
The non-strict version is A264401.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A257993 gives the least gap of the partition with Heinz number n.
A339564 counts factorizations with a selected factor.
A342050 ranks partitions with even least gap.
A342051 ranks partitions with odd least gap.
Cf. A003242, A083710, A083711, A097986, A098743, A098965, A130689, A200745, A341450, A343347, A343377.

mingap[q_]:=Min@@Complement[Range[If[q=={},0,Max[q]]+1],q];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&mingap[#]==k&]],{n,0,15},{k,Round[Sqrt[2*(n+1)]]}]

A294141 Number of partitions of n into odd parts that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 3, 0, 5, 5, 2, 7, 5, 1, 11, 14, 6, 9, 20, 5, 18, 33, 5, 43, 50, 12, 58, 17, 16, 91, 96, 26, 58, 146, 23, 183, 148, 11, 241, 285, 88, 187, 152, 98, 367, 537, 87, 177, 376, 179, 857, 983, 77, 1195, 1274, 79, 1596, 468, 290, 2117, 1887, 549, 460, 3064, 440

Offset: 0

Ilya Gutkovskiy, Oct 23 2017

nonn

			a(13) = 2 because we have [7, 3, 3] and [5, 5, 3].

Index entries for sequences related to partitions

Cf. A000009, A098743, A128515, A171565, A200745, A294142.

Table[SeriesCoefficient[Product[1/(1 - Boole[Mod[n, k] > 0 && OddQ[k]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 72}]

A294142 Number of partitions of n into distinct odd parts that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 1, 2, 2, 3, 0, 4, 4, 4, 2, 6, 1, 6, 8, 4, 10, 12, 4, 12, 5, 7, 17, 17, 8, 14, 24, 9, 29, 24, 4, 33, 40, 25, 29, 28, 23, 45, 63, 23, 30, 52, 37, 84, 99, 26, 113, 112, 23, 143, 60, 57, 173, 143, 89, 70, 226, 87, 256, 256, 53, 245, 135, 127, 378, 233

Offset: 0

Ilya Gutkovskiy, Oct 23 2017

nonn

			a(14) = 2 because we have [11, 3] and [9, 5].

Index entries for sequences related to partitions

Cf. A000700, A098743, A171565, A200745, A209402, A294141.

Table[SeriesCoefficient[Product[1 + Boole[Mod[n, k] > 0 && OddQ[k]] x^k, {k, 1, n}], {x, 0, n}], {n, 0, 80}]

A298262 Number of integer partitions of n using relatively prime non-divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 1, 3, 2, 13, 1, 23, 7, 10, 8, 65, 5, 104, 11, 53, 53, 252, 8, 244, 124, 203, 67, 846, 22, 1237, 157, 636, 569, 1074, 51, 3659, 1140, 1827, 221, 7244, 236, 10086, 1162, 1844, 4169, 19195, 225, 17657, 2997

Offset: 1

Gus Wiseman, Jan 15 2018

nonn

			The a(11) = 13 partitions: (65), (74), (83), (92), (443), (533), (542), (632), (722), (3332), (4322), (5222), (32222).
The a(14) = 7 partitions: (9 5), (11 3), (5 5 4), (6 5 3), (8 3 3), (4 4 3 3), (5 3 3 3).

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000041, A000837, A002577, A002865, A018818, A018819, A049820, A098743, A100953, A200745, A275870, A281116.

Table[Length[Select[IntegerPartitions[n],And[GCD@@#===1,!Or@@(Divisible[n,#]&/@#)]&]],{n,50}]

\\ here b(n) is A098743.
b(n)={polcoef(1/prod(k=1, n, if(n%k, 1 - x^k, 1) + O(x*x^n)), n)}
a(n)={sumdiv(n, d, moebius(d)*b(n/d))} \\ Andrew Howroyd, Aug 29 2018

a(n) = Sum_{d|n} mu(n/d) * A098743(d).

A332001 Number of compositions (ordered partitions) of n into distinct parts that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 0, 4, 2, 4, 4, 20, 2, 34, 14, 20, 14, 146, 8, 244, 22, 140, 202, 956, 16, 782, 596, 752, 216, 5786, 82, 10108, 640, 4016, 5200, 6028, 218, 53674, 14570, 19004, 980, 152810, 1786, 245884, 13588, 16534, 108382, 719156, 1494, 532532, 54316

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

			a(9) = 4 because we have [7, 2], [5, 4], [4, 5] and [2, 7].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to compositions

Cf. A018818, A032020, A033630, A098743, A100346, A200745, A300702, A331927, A331979.

a:= proc(n) local b, l; l, b:= numtheory[divisors](n),
      proc(m, i, p) option remember; `if`(m=0, p!, `if`(i<2, 0,
        b(m, i-1, p)+`if`(i>m or i in l, 0, b(m-i, i-1, p+1))))
      end; forget(b): b(n, n-1, 0)
    end:
seq(a(n), n=0..63);  # Alois P. Heinz, Feb 04 2020

a[n_] := Module[{b, l = Divisors[n]}, b[m_, i_, p_] := b[m, i, p] = If[m == 0, p!, If[i < 2, 0, b[m, i - 1, p] + If[i > m || MemberQ[l, i], 0, b[m - i, i - 1, p + 1]]]]; b[n, n - 1, 0]];
a /@ Range[0, 63] (* Jean-François Alcover, Nov 30 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A300702 Number of compositions (ordered partitions) of n into parts that do not divide n.

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Maple

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A294266 Number of partitions of n into distinct squares that do not divide n.

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Maple

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A300584 Number of partitions of n into distinct prime power parts (not including 1) that do not divide n.

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Mathematica

A339660 Number of strict integer partitions of n with no 1's and a part divisible by all the other parts.

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Mathematica

A300586 Number of partitions of n into distinct squarefree parts that do not divide n.

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Mathematica

A343348 Irregular triangle read by rows where T(n,k) is the number of strict integer partitions of n with least gap k.

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Mathematica

A294141 Number of partitions of n into odd parts that do not divide n.

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Mathematica

A294142 Number of partitions of n into distinct odd parts that do not divide n.

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Mathematica

A298262 Number of integer partitions of n using relatively prime non-divisors of n.

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