A032022 -id:A032022 - cp's OEIS frontend

A331844 Number of compositions (ordered partitions) of n into distinct squares.

1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 3, 8, 0, 0, 8, 30, 0, 0, 0, 2, 6, 1, 2, 6, 24, 2, 8, 6, 0, 0, 8, 30, 0, 0, 7, 32, 24, 2, 8, 30, 120, 6, 24, 2, 6, 0, 8, 36, 24, 1, 34, 150, 0, 2, 12, 30, 24, 0, 2, 38, 150, 0, 12, 78, 144, 2

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

			a(14) = 6 because we have [9,4,1], [9,1,4], [4,9,1], [4,1,9], [1,9,4] and [1,4,9].

Cf. A000290, A006456, A032020, A032021, A032022, A033461, A218396, A219107, A331843, A331845, A331846, A331847.

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)*(2*i+1)/6n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..82);  # Alois P. Heinz, Jan 30 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n, Sqrt[n] // Floor, 0];
a /@ Range[0, 82] (* Jean-François Alcover, Oct 29 2020, after Alois P. Heinz *)

A331843 Number of compositions (ordered partitions) of n into distinct triangular numbers.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 0, 2, 7, 2, 0, 2, 6, 1, 4, 6, 2, 12, 24, 3, 8, 0, 8, 32, 6, 2, 13, 26, 6, 34, 36, 0, 32, 150, 3, 20, 50, 14, 54, 126, 32, 32, 12, 55, 160, 78, 122, 44, 174, 4, 72, 294, 36, 201, 896, 128, 62, 180, 176, 164, 198, 852, 110, 320, 159, 212, 414

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

			a(10) = 7 because we have [10], [6, 3, 1], [6, 1, 3], [3, 6, 1], [3, 1, 6], [1, 6, 3] and [1, 3, 6].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Index entries for sequences related to compositions

Cf. A000217, A023361, A024940, A032020, A032021, A032022, A218396, A219107, A331844, A331845, A331846, A331847.

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), 1+h(n-1), h(n-1)))
    end:
b:= proc(n, i, p) option remember; (t->
      `if`(t*(i+2)/3n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
    end:
a:= n-> b(n, h(n), 0):
seq(a(n), n=0..73);  # Alois P. Heinz, Jan 31 2020

h[n_] := h[n] = If[n<1, 0, If[IntegerQ @ Sqrt[8n+1], 1 + h[n-1], h[n-1]]];
b[n_, i_, p_] := b[n, i, p] = Function[t, If[t (i + 2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t>n, 0, b[n - t, i - 1, p + 1]]]]][(i(i + 1)/2)];
a[n_] := b[n, h[n], 0];
a /@ Range[0, 73] (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

A337451 Number of relatively prime strict compositions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 10, 8, 20, 14, 34, 52, 72, 90, 146, 172, 244, 390, 502, 680, 956, 1218, 1686, 2104, 3436, 4078, 5786, 7200, 10108, 12626, 17346, 20876, 32836, 38686, 53674, 67144, 91528, 113426, 152810, 189124, 245884, 343350, 428494, 552548, 719156

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

			The a(5) = 2 through a(10) = 8 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)    (3,7)
  (3,2)     (3,4)  (5,3)  (4,5)    (7,3)
            (4,3)         (5,4)    (2,3,5)
            (5,2)         (7,2)    (2,5,3)
                          (2,3,4)  (3,2,5)
                          (2,4,3)  (3,5,2)
                          (3,2,4)  (5,2,3)
                          (3,4,2)  (5,3,2)
                          (4,2,3)
                          (4,3,2)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

A032022 does not require relative primality.
A302698 is the unordered non-strict version.
A332004 is the version allowing 1's.
A337450 is the non-strict version.
A337452 is the unordered version.
A000837 counts relatively prime partitions.
A032020 counts strict compositions.
A078374 counts strict relatively prime partitions.
A002865 counts partitions with no 1's.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337462 counts pairwise coprime compositions.
A337561 counts strict pairwise coprime compositions.
Cf. A000010, A007359, A101268, A178472, A216652, A289509, A337562, A337563.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

A339101 Number of compositions (ordered partitions) of n into distinct parts >= 3.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 3, 3, 5, 5, 7, 13, 15, 21, 29, 35, 43, 79, 87, 123, 161, 221, 259, 349, 531, 645, 857, 1115, 1471, 1903, 2403, 2979, 4493, 5357, 7135, 9013, 11919, 14925, 19685, 23939, 30667, 42679, 52215, 67035, 86009, 109541, 137923, 177493, 222027, 277749

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Index entries for sequences related to compositions

Cf. A008483, A025148, A032020, A032022, A078012, A339102, A339103, A339104.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-2)*(i+3)/2 b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 23 2020

nmax = 50; CoefficientList[Series[Sum[k! x^(k (k + 5)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(k + 5)/2) / Product_{j=1..k} (1 - x^j).

A339163 Number of compositions (ordered partitions) of n into distinct parts, the least being 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 2, 2, 2, 8, 8, 14, 14, 20, 44, 50, 74, 104, 128, 158, 326, 356, 524, 698, 986, 1160, 1592, 2606, 3158, 4316, 5708, 7706, 10082, 12920, 16136, 25718, 30614, 41756, 53396, 71978, 91058, 122144, 149384, 193670, 279614, 342860, 447764, 581234

Offset: 0

Ilya Gutkovskiy, Nov 25 2020

nonn

			a(9) = 8 because we have [7, 2], [4, 3, 2], [4, 2, 3], [3, 4, 2], [3, 2, 4], [2, 7], [2, 4, 3] and [2, 3, 4].

Index entries for sequences related to compositions

Cf. A002865, A032022, A096749, A339162, A339164, A339165, A339166.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-2)*(i+3)/2 `if`(n<2, 0, b(n-2$2, 1)):
seq(a(n), n=0..55);  # Alois P. Heinz, Nov 25 2020

nmax = 47; CoefficientList[Series[Sum[k! x^(k (k + 3)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k! * x^(k*(k + 3)/2) / Product_{j=1..k-1} (1 - x^j).

A331845 Number of compositions (ordered partitions) of n into distinct cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 6, 24

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

			a(36) = 6 because we have [27,8,1], [27,1,8], [8,27,1], [8,1,27], [1,27,8] and [1,8,27].

Cf. A000578, A023358, A032020, A032021, A032022, A218396, A219107, A279329, A331843, A331844, A331846, A331847.

b:= proc(n, i, p) option remember;
      `if`((i*(i+1)/2)^2n, 0, b(n-i^3, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, iroot(n, 3), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 30 2020

b[n_, i_, p_] := b[n, i, p] = If[(i(i+1)/2)^2 < n, 0, If[n == 0, p!, If[i^3 > n, 0, b[n-i^3, i-1, p+1]] + b[n, i-1, p]]];
a[n_] := b[n, Floor[n^(1/3)], 0];
a /@ Range[0, 100] (* Jean-François Alcover, Oct 31 2020, after Alois P. Heinz *)

A339102 Number of compositions (ordered partitions) of n into distinct parts >= 4.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 15, 15, 23, 29, 37, 43, 57, 87, 101, 137, 175, 235, 279, 363, 431, 665, 757, 1015, 1257, 1683, 2069, 2645, 3199, 4063, 5607, 6759, 8591, 10877, 13837, 17251, 22185, 26871, 33773, 41273, 56047, 66499, 85647, 104811

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(9) = 3 because we have [9], [5, 4] and [4, 5].

Index entries for sequences related to compositions

Cf. A008484, A017898, A025149, A032020, A032022, A339101, A339103, A339104.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-3)*(i+4)/2 b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 23 2020

nmax = 52; CoefficientList[Series[Sum[k! x^(k (k + 7)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(k + 7)/2) / Product_{j=1..k} (1 - x^j).

A339103 Number of compositions (ordered partitions) of n into distinct parts >= 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 15, 17, 23, 31, 37, 45, 57, 65, 101, 115, 151, 189, 255, 293, 383, 451, 565, 777, 921, 1157, 1469, 1855, 2311, 2865, 3495, 4313, 5231, 7063, 8269, 10509, 12849, 16217, 19829, 25171, 30031, 37485, 45183

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

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

Index entries for sequences related to compositions

Cf. A017899, A025150, A032020, A032022, A185325, A339101, A339102, A339104.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-4)*(i+5)/2 b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 23 2020

nmax = 54; CoefficientList[Series[Sum[k! x^(k (k + 9)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(k + 9)/2) / Product_{j=1..k} (1 - x^j).

A339104 Number of compositions (ordered partitions) of n into distinct parts >= 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 17, 17, 25, 31, 39, 45, 59, 65, 79, 115, 129, 165, 209, 269, 313, 403, 471, 585, 683, 941, 1063, 1375, 1641, 2097, 2537, 3161, 3745, 4663, 5535, 6741, 8627, 10241, 12535, 15307, 18849, 22869, 28409

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(13) = 3 because we have [13], [7, 6] and [6, 7].

Index entries for sequences related to compositions

Cf. A017900, A025151, A032020, A032022, A185326, A339101, A339102, A339103.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-5)*(i+6)/2 b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 23 2020

nmax = 57; CoefficientList[Series[Sum[k! x^(k (k + 11)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(k + 11)/2) / Product_{j=1..k} (1 - x^j).

A331846 Number of compositions (ordered partitions) of n into distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 9, 5, 12, 16, 21, 41, 42, 49, 59, 79, 130, 231, 230, 295, 226, 495, 609, 699, 1472, 1042, 1377, 2308, 2982, 3425, 3879, 4877, 7156, 7189, 13531, 14797, 13570, 19551, 27667, 30327, 36382, 47519, 60783, 70561, 78330, 136988, 121659, 174851

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

			a(7) = 5 because we have [7], [6, 1], [5, 2], [2, 5] and [1, 6].

Index entries for sequences related to compositions

Cf. A005117, A032020, A032021, A032022, A087188, A218396, A219107, A280194, A331843, A331844, A331845, A331847.

cp's OEIS Frontend

A331844 Number of compositions (ordered partitions) of n into distinct squares.

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A331843 Number of compositions (ordered partitions) of n into distinct triangular numbers.

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A337451 Number of relatively prime strict compositions of n with no 1's.

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A339101 Number of compositions (ordered partitions) of n into distinct parts >= 3.

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A339163 Number of compositions (ordered partitions) of n into distinct parts, the least being 2.

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A331845 Number of compositions (ordered partitions) of n into distinct cubes.

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A339102 Number of compositions (ordered partitions) of n into distinct parts >= 4.

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A339103 Number of compositions (ordered partitions) of n into distinct parts >= 5.

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