A224344 -id:A224344 - cp's OEIS frontend

A052284 Number of compositions of n into nonprime numbers.

1, 1, 1, 1, 2, 3, 5, 7, 11, 17, 27, 40, 61, 92, 142, 217, 333, 506, 774, 1181, 1807, 2758, 4215, 6434, 9833, 15019, 22948, 35047, 53541, 81780, 124936, 190841, 291532, 445320, 680274, 1039155, 1587405, 2424849, 3704148, 5658321, 8643530

Offset: 0

Robert G. Wilson v, May 16 2002

nonn

Starting at n=1, appears to be row sums of triangle A157424. - Gary W. Adamson & Mats Granvik, Feb 28 2009

			a(6) = 5 because 1+1+1+1+1+1 = 1+1+4 = 1+4+1 = 4+1+1 = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..5437 (first 501 terms from T. D. Noe)

Cf. A002095 (Number of partitions of n into nonprime parts).
Cf. A000041 & A023360.
Column k=0 of A224344.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(isprime(j), 0, a(n-j)), j=1..n))
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Aug 06 2019

nn=50; np=Select[Range[nn], !PrimeQ[ # ] &]; CoefficientList[Series[1/(1-Sum[x^k, {k, np}]), {x, 0, nn}], x] (* T. D. Noe, Aug 20 2010 *)

G.f.: 1/( 1 - (Sum_{m nonprime} x^m) ).

Definition and g.f. corrected by N. J. A. Sloane, Aug 19 2010, who thanks Vladimir Kruchinin for pointing out the errors.

A121303 Triangle read by rows: T(n,k) is the number of compositions of n into k primes (i.e., ordered sequences of k primes having sum n; n>=2, k>=1).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 2, 3, 0, 2, 3, 1, 0, 2, 4, 4, 0, 3, 6, 6, 1, 1, 0, 6, 8, 5, 0, 2, 9, 13, 10, 1, 1, 2, 6, 16, 15, 6, 0, 3, 6, 22, 25, 15, 1, 0, 2, 10, 24, 36, 26, 7, 0, 4, 9, 22, 50, 45, 21, 1, 1, 0, 12, 32, 65, 72, 42, 8, 0, 4, 12, 34, 70, 106, 77, 28, 1, 1, 2, 12, 40, 90, 150

Offset: 2

Emeric Deutsch, Aug 06 2006

Row n has floor(n/2) terms.
Sum of terms in row n = A023360(n).
T(n,1) = A010051(n) (characteristic function of primes); T(n,2) = A073610(n); T(n,3) = A098238(n).
Sum_{k=1..floor(n/2)} k*T(n,k) = A121304(n).

			T(9,3) = 4 because we have [2,2,5], [2,5,2], [5,2,2] and [3,3,3].
Triangle starts:
  1;
  1;
  0, 1;
  1, 2;
  0, 1, 1;
  1, 2, 3;
  0, 2, 3, 1;
  0, 2, 4, 4;
  ...

Alois P. Heinz, Rows n = 2..200, flattened

Cf. A010051, A023360, A073610, A098238, A121304, A224344.

T(n^2,n) gives A341459.

G:=1/(1-t*sum(z^ithprime(i),i=1..30))-1: Gser:=simplify(series(G,z=0,25)): for n from 2 to 21 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 2 to 21 do seq(coeff(P[n],t,j),j=1..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
with(numtheory):
b:= proc(n) option remember; local j; if n=0 then [1]
      else []; for j to pi(n) do zip((x, y)->x+y, %,
      [0, b(n-ithprime(j))[]], 0) od; % fi
    end:
T:= n-> subsop(1=NULL, b(n))[]:
seq(T(n), n=2..20);  # Alois P. Heinz, May 23 2013

nn=20;a[x_]:=Sum[x^Prime[n],{n,1,nn}];CoefficientList[Series[1/(1-y a[x]),{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Nov 08 2013 *)

G.f.: 1/(1 - t*Sum_{i>=1} z^prime(i)).

A102291 Total number of prime parts in all compositions of n.

Original entry on oeis.org

0, 0, 1, 3, 7, 18, 42, 98, 222, 497, 1100, 2413, 5250, 11350, 24398, 52193, 111180, 235949, 499074, 1052502, 2213710, 4644833, 9724492, 20318637, 42376578, 88231765, 183420748, 380755932, 789340736, 1634339217, 3379993922, 6982618822, 14410499598, 29711523105

Offset: 0

Vladeta Jovovic, Feb 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..3313

Cf. A037032, A224344, A336632.

a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
      `if`(isprime(j), ceil(2^(n-j-1)), 0), j=1..n))
    end:
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 06 2019

a[n_] := a[n] = If[n==0, 0, Sum[a[n-j] + If[PrimeQ[j], Ceiling[2^(n-j-1)], 0], {j, 1, n}]];
a /@ Range[0, 33] (* Jean-François Alcover, Oct 30 2020, after Alois P. Heinz *)

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

a(n) = Sum_{k=1..floor(n/2)} k * A224344(n,k). - Alois P. Heinz, Aug 06 2019

More terms from Joshua Zucker, May 10 2006

A222656 Number T(n,k) of partitions of n using exactly k primes; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 2, 3, 4, 3, 1, 3, 6, 4, 2, 5, 7, 6, 3, 1, 6, 9, 8, 5, 2, 8, 11, 12, 7, 3, 1, 8, 17, 14, 10, 5, 2, 12, 20, 19, 14, 8, 3, 1, 13, 26, 25, 19, 11, 5, 2, 17, 31, 35, 24, 16, 8, 3, 1, 19, 41, 42, 34, 21, 12, 5, 2, 26, 47, 56, 44, 29

Offset: 0

Alois P. Heinz, May 29 2013

			T(6,0) = 3: [6], [4,1,1], [1,1,1,1,1,1].
T(6,1) = 4: [5,1], [4,2], [3,1,1,1], [2,1,1,1,1].
T(6,2) = 3: [3,3], [3,2,1], [2,2,1,1].
T(6,3) = 1: [2,2,2].
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  2;
  2,  2,  1;
  2,  3,  2;
  3,  4,  3,  1;
  3,  6,  4,  2;
  5,  7,  6,  3, 1;
  6,  9,  8,  5, 2;
  8, 11, 12,  7, 3, 1;
  8, 17, 14, 10, 5, 2;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Column k=0 gives: A002095.
Row sums give: A000041.
Cf. A000040, A004526, A224344.

b:= proc(n, i) option remember; local j; if n=0 then 1 elif i<1 then 0
      else []; for j from 0 to n/i do zip((x, y)->x+y, %,
      [`if`(isprime(i), 0$j, NULL), b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> b(n$2):
seq(T(n), n=0..16);

zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, Thread[f[PadRight[x, m, z], PadRight[y, m, z]]]]; b[n_, i_] := b[n, i] = Module[{j, pc}, Which[n == 0, {1}, i<1, {0}, True, pc = {}; For[j = 0, j <= n/i, j++, pc = zip[Plus, pc, Join[If[PrimeQ[i], Array[0&, j], {}], b[n-i*j, i-1]], 0]]; pc]]; T[n_] := b[n, n]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)

Sum_{k=1..floor(n/2)} k * T(n,k) = A037032(n).

G.f.: G(t,x) = Product_{i>=1} (1 - x^prime(i))/((1 - x^i)*(1 - t*x^prime(i))). - Emeric Deutsch, Nov 11 2015

A299168 Number of ordered ways of writing n-th prime number as a sum of n primes.

Original entry on oeis.org

1, 0, 0, 0, 5, 6, 42, 64, 387, 5480, 10461, 113256, 507390, 1071084, 4882635, 44984560, 382362589, 891350154, 7469477771, 33066211100, 78673599501, 649785780710, 2884039365010, 22986956007816, 306912836483025, 1361558306986280, 3519406658042964

Offset: 1

Ilya Gutkovskiy, Feb 04 2018

nonn

			a(5) = 5 because fifth prime number is 11 and we have [3, 2, 2, 2, 2], [2, 3, 2, 2, 2], [2, 2, 3, 2, 2], [2, 2, 2, 3, 2] and [2, 2, 2, 2, 3].

Alois P. Heinz, Table of n, a(n) for n = 1..400

Cf. A000040, A000586, A000607, A023360, A056768, A070215, A073610, A098238, A117278, A121303, A219180, A224344, A259254, A265112, A341459.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(isprime(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(ithprime(n), n):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 13 2021

Table[SeriesCoefficient[Sum[x^Prime[k], {k, 1, n}]^n, {x, 0, Prime[n]}], {n, 1, 27}]

a(n) = [x^prime(n)] (Sum_{k>=1} x^prime(k))^n.

cp's OEIS Frontend

A052284 Number of compositions of n into nonprime numbers.

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A121303 Triangle read by rows: T(n,k) is the number of compositions of n into k primes (i.e., ordered sequences of k primes having sum n; n>=2, k>=1).

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A102291 Total number of prime parts in all compositions of n.

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A222656 Number T(n,k) of partitions of n using exactly k primes; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

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A299168 Number of ordered ways of writing n-th prime number as a sum of n primes.

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