A052284 -id:A052284 - cp's OEIS frontend

A347788 Number of compositions (ordered partitions) of n into at most 2 nonprime parts.

1, 1, 1, 0, 1, 2, 1, 2, 2, 3, 5, 2, 4, 4, 5, 5, 8, 4, 8, 6, 8, 7, 11, 6, 12, 9, 13, 9, 14, 10, 16, 12, 14, 13, 19, 13, 22, 14, 17, 17, 22, 16, 24, 18, 22, 19, 25, 18, 28, 21, 28, 21, 28, 22, 32, 25, 30, 25, 33, 26, 38, 28, 31, 29, 38, 29, 42, 30, 34, 33, 42

Offset: 0

Ilya Gutkovskiy, Sep 13 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A005171, A018252, A052284, A076608, A347796, A347797, A347798, A347799, A358638.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,2,Select[Range@n,!PrimeQ@#&]],1],{n,0,70}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A347788(n) = if(n<2,1,!isprime(n)+sum(k=1,n-1,!(isprime(k)+isprime(n-k)))); \\ Antti Karttunen, Nov 25 2022

A347796 Number of compositions (ordered partitions) of n into at most 3 nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 2, 5, 6, 8, 11, 8, 13, 17, 17, 23, 25, 27, 33, 38, 37, 50, 48, 58, 63, 73, 70, 89, 82, 107, 102, 122, 112, 148, 136, 164, 161, 185, 173, 223, 196, 241, 231, 268, 254, 304, 273, 332, 318, 364, 348, 403, 364, 444, 415, 477, 448, 525, 479, 567

Offset: 0

Ilya Gutkovskiy, Sep 14 2021

nonn

Cf. A018252, A052284, A341480, A347788, A347797, A347798, A347799.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,3,Select[Range@n,!PrimeQ@#&]],1],{n,0,60}] (* Giorgos Kalogeropoulos, Sep 14 2021 *)

A347797 Number of compositions (ordered partitions) of n into at most 4 nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 6, 5, 10, 14, 15, 24, 21, 35, 45, 48, 65, 77, 89, 114, 129, 148, 184, 205, 239, 285, 310, 361, 410, 459, 522, 593, 636, 740, 804, 911, 969, 1123, 1169, 1350, 1428, 1595, 1687, 1926, 1974, 2270, 2325, 2611, 2726, 3064, 3120, 3547, 3596, 4012, 4155

Offset: 0

Ilya Gutkovskiy, Sep 14 2021

nonn

Cf. A018252, A052284, A341481, A347788, A347796, A347798, A347799.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,4,Select[Range@n,!PrimeQ@#&]],1],{n,0,55}] (* Giorgos Kalogeropoulos, Sep 14 2021 *)

A347798 Number of compositions (ordered partitions) of n into at most 5 nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 10, 10, 19, 25, 29, 46, 50, 75, 103, 110, 162, 194, 240, 309, 368, 444, 565, 654, 795, 960, 1106, 1325, 1560, 1792, 2118, 2436, 2785, 3244, 3691, 4194, 4783, 5419, 6121, 6893, 7780, 8617, 9766, 10790, 12060, 13340, 14851, 16231, 18210

Offset: 0

Ilya Gutkovskiy, Sep 14 2021

nonn

Cf. A018252, A052284, A341482, A347788, A347796, A347797, A347799.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,5,Select[Range@n,!PrimeQ@#&]],1],{n,0,50}] (* Giorgos Kalogeropoulos, Sep 14 2021 *)

A347799 Number of compositions (ordered partitions) of n into at most 6 nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 6, 10, 16, 19, 31, 44, 52, 86, 101, 148, 206, 237, 350, 441, 551, 743, 900, 1151, 1470, 1782, 2216, 2762, 3287, 4072, 4894, 5835, 7052, 8362, 9856, 11758, 13710, 16066, 18791, 21799, 25271, 29192, 33583, 38485, 44178, 50304

Offset: 0

Ilya Gutkovskiy, Sep 14 2021

nonn

Cf. A018252, A052284, A341483, A347788, A347796, A347797, A347798.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,6,Select[Range@n,!PrimeQ@#&]],1],{n,0,46}] (* Giorgos Kalogeropoulos, Sep 14 2021 *)

A353429 Number of integer compositions of n with all prime parts and all prime run-lengths.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 4, 0, 2, 2, 5, 4, 9, 1, 5, 12, 20, 11, 19, 18, 31, 43, 54, 37, 63, 95, 121, 124, 154, 178, 261, 353, 393, 417, 565, 770, 952, 1138, 1326, 1647, 2186, 2824, 3261, 3917, 4941, 6423, 7935, 9719, 11554, 14557, 18536, 23380, 27985

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(13) = 2 through a(16) = 9 compositions:
  (22333)  (77)       (555)     (3355)
  (33322)  (2255)     (33333)   (5533)
           (5522)     (222333)  (22255)
           (223322)   (333222)  (55222)
           (2222222)            (332233)
                                (2222233)
                                (2223322)
                                (2233222)
                                (3322222)

Alois P. Heinz, Table of n, a(n) for n = 0..4000

The first condition only is A023360, partitions A000607.
For partitions we have A351982, only run-lens A100405, only parts A008483.
The second condition only is A353401, partitions A055923.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A052284 counts compositions into nonprimes, partitions A002095.
A106356 counts compositions by number of adjacent equal parts.
A114901 counts compositions with no runs of length 1, ranked by A353427.
A329738 counts uniform compositions, partitions A047966.
Cf. A005811, A128695, A137200, A167606, A333755, A335464, A353428.

b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h and isprime(i),
      add(`if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=2..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, May 18 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,15}]

a(26)-a(56) from Alois P. Heinz, May 18 2022

A157423 Triangle read by rows, T(n,k) = 0 if (n-k+1) is prime, else 1.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 28 2009

Row sums = A062298: (1, 1, 1, 2, 2, 3, 3, 4, 5, 6,...). Eigensequence of the triangle = A052284: (1, 1, 1, 2, 3, 5, 7, 11, 17, 27,...).

			First few rows of the triangle =
1;
0, 1;
0, 0, 1;
1, 0, 0, 1;
0, 1, 0, 0, 1;
1, 0, 1, 0, 0, 1;
0, 1, 0, 1, 0, 0, 1;
1, 0, 1, 0, 1, 0, 0, 1;
1, 1, 0, 1, 0, 1, 0, 0, 1;
1, 1, 1, 0, 1, 0, 1, 0, 0, 1;
0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1;
1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1;
0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1;
1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1;
...
Example: T(6,4) = 0 since (6 - 4 + 1) = 3, prime.

Cf. A005171, A062298, A052284, A157424

Table[If[PrimeQ[n-k+1],0,1],{n,15},{k,n}]//Flatten (* Harvey P. Dale, Jul 19 2016 *)

Triangle read by rows, T(n,k) = 0 if (n-k+1) is prime, else 1. By columns, A005171 in every column: where A005171(k) = 0 if k is prime.

A276421 Number of palindromic compositions of n into nonprime numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 5, 3, 7, 4, 11, 6, 16, 9, 25, 14, 38, 21, 59, 34, 89, 50, 137, 77, 208, 117, 319, 180, 486, 273, 744, 420, 1134, 639, 1735, 977, 2648, 1491, 4048, 2281, 6180, 3480, 9444, 5321, 14421, 8122, 22035, 12412, 33655, 18957, 51417, 28966

Offset: 0

Emeric Deutsch, Sep 03 2016

nonn

			a(6) = 3 because we have [6], [1,4,1], and [1,1,1,1,1,1].
a(10) = 7 because we have [10], [1,8,1], [1,1,6,1,1], [1,4,4,1], [4,1,1,4], [1,1,1,4,1,1,1], and [1^{10}].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Cf. A016116, A052284, A276420.

F:=sum(z^j,j=1..229)-(sum(z^ithprime(k),k=1..50)): g:=(1+F)/(1-subs(z = z^2, F)): gser:=series(g,z=0,53): seq(coeff(gser,z,n),n=0..50);
# second Maple program:
a:= proc(n) option remember; `if`(isprime(n), 0, 1)+
      add(`if`(isprime(j), 0, a(n-2*j)), j=1..n/2)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 03 2016

a[n_] := a[n] = If[PrimeQ[n], 0, 1] + Sum[If[PrimeQ[j], 0, a[n-2j]], {j, 1, n/2}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)

G.f.: g(z)=(1+F(z))/(1-F(z^2)), where F(z)=Sum_{m nonprime} z^m = z + z^4 + z^6 + z^8 + z^9 + z^10 + z^12 + ... is the g.f. of A005171.

A280544 Expansion of 1/(1 - Sum_{k>=2} (1 - floor(2/d(k)))*x^k), where d(k) is the number of divisors (A000005).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 3, 0, 5, 2, 8, 3, 13, 5, 22, 10, 34, 18, 58, 31, 94, 57, 153, 99, 254, 172, 417, 302, 685, 523, 1136, 901, 1872, 1557, 3097, 2673, 5133, 4577, 8505, 7843, 14109, 13380, 23440, 22816, 38953, 38855, 64789, 66053, 107871, 112190, 179664, 190369, 299478, 322683, 499501, 546548

Offset: 0

Ilya Gutkovskiy, Jan 05 2017

nonn

Number of compositions (ordered partitions) of n into composite parts (A002808).

			a(10) = 3 because we have [10], [6, 4] and [4, 6].

Eric Weisstein's World of Mathematics, Composite Number
Index entries for sequences related to compositions

Cf. A000005, A002808, A023360, A023895, A052284.

nmax = 59; CoefficientList[Series[1/(1 - Sum[(1 - Floor[2/DivisorSigma[0, k]]) x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

x='x+O('x^60); Vec(1/(1 - sum(k=2, 59, (1 - 2\numdiv(k))*x^k))) \\ Indranil Ghosh, Apr 03 2017

G.f.: 1/(1 - Sum_{k>=2} (1 - floor(2/d(k)))*x^k).

A331917 Number of compositions (ordered partitions) of n into distinct nonprime parts.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 2, 1, 3, 5, 8, 3, 10, 11, 17, 13, 16, 19, 54, 49, 55, 59, 90, 89, 129, 127, 183, 307, 358, 351, 456, 553, 649, 889, 1015, 1143, 1490, 2219, 1913, 3021, 3394, 4241, 4944, 6663, 6859, 9337, 9522, 12123, 14895, 22425, 18849, 28341, 31468, 41533

Offset: 0

Ilya Gutkovskiy, Feb 01 2020

nonn

			a(10) = 5 because we have [10], [9, 1], [6, 4], [4, 6] and [1, 9].

Index entries for sequences related to compositions

Cf. A002095, A018252, A052284, A096258, A219107.

cp's OEIS Frontend

A347788 Number of compositions (ordered partitions) of n into at most 2 nonprime parts.

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A347796 Number of compositions (ordered partitions) of n into at most 3 nonprime parts.

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A347797 Number of compositions (ordered partitions) of n into at most 4 nonprime parts.

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A347798 Number of compositions (ordered partitions) of n into at most 5 nonprime parts.

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A347799 Number of compositions (ordered partitions) of n into at most 6 nonprime parts.

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A353429 Number of integer compositions of n with all prime parts and all prime run-lengths.

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A157423 Triangle read by rows, T(n,k) = 0 if (n-k+1) is prime, else 1.

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A276421 Number of palindromic compositions of n into nonprime numbers.

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A280544 Expansion of 1/(1 - Sum_{k>=2} (1 - floor(2/d(k)))*x^k), where d(k) is the number of divisors (A000005).

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