A023360 -id:A023360 - cp's OEIS frontend

A265112 a(n) = A023360(A000040(n)): number of compositions of prime(n) into prime parts.

1, 1, 3, 6, 20, 46, 232, 501, 2352, 24442, 53243, 550863, 2616338, 5701553, 27077005, 280237217, 2900328380, 6320545915, 65414893802, 310664269401, 677015556295, 7006815193063, 33276323565116, 344395408399372, 7767597342090622, 36889382062795742

Offset: 1

Bob Selcoe, Dec 01 2015

			prime(4) = 7; a(4) = A023360(7) = 6 because there are 6 compositions of 7 into prime parts {2,3,5,7}: {7}, {5+2}, {3+2+2}, {2+5}, {2+3+2} and {2+2+3}.

Robert Israel, Table of n, a(n) for n = 1..768

Cf. A000040 (prime numbers), A023360.

N:= 1000: # to get a(1) to a(A000720(N))
Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
M:= nops(Primes);
F:= proc(x)
option remember;
local k;
add(procname(x-Primes[k]),k=1..numtheory:-pi(x));
end proc:
F(0):= 1:
seq(F(Primes[n]),n=1..M); # Robert Israel, Dec 02 2015

Needs["Combinatorica`"]; Table[Length@ Flatten[Permutations[#, {Length@ #}] & /@ Select[Combinatorica`Partitions@ Prime@ n, AllTrue[#, PrimeQ] &], 1], {n, 14}] (* Version 10, slow, or *)
lim = 101; t = Rest@ CoefficientList[Series[1/(1 - Sum[x^Prime[i], {i, 1, PrimePi@ lim}]), {x, 0, lim}], x]; t[[#]] &@ Prime@ Range@ PrimePi@ lim
(* Michael De Vlieger, Dec 01 2015, after David W. Wilson at A023360 *)

A052284 Number of compositions of n into nonprime numbers.

Original entry on oeis.org

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.

A280917 Expansion of 1/(1 - x - Sum_{k>=1} x^prime(k)).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 50, 95, 180, 343, 652, 1240, 2359, 4486, 8532, 16227, 30862, 58697, 111636, 212321, 403814, 768015, 1460691, 2778094, 5283667, 10049027, 19112282, 36349721, 69133673, 131485594, 250072951, 475614693, 904573387, 1720411555, 3272057256, 6223138101, 11835809946, 22510571803, 42812941849

Offset: 0

Ilya Gutkovskiy, Jan 10 2017

nonn

Number of compositions (ordered partitions) of n into prime parts (1 included) (A008578).

			a(4) = 7 because we have [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].

Indranil Ghosh, Table of n, a(n) for n = 0..99
Index entries for sequences related to compositions

Cf. A000607, A002124, A008578, A023360, A034891, A036497.

nmax = 39; CoefficientList[Series[1/(1 - x - Sum[x^Prime[k], {k, 1, nmax}]), {x, 0, nmax}], x]

Vec(1 / (1 - x - sum(k=1, 100,  x^prime(k))) + O(x^100)) \\ Indranil Ghosh, Mar 09 2017

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

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)).

A280195 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(k))*x^k), where omega(k) is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 4, 8, 11, 19, 28, 47, 72, 116, 182, 289, 460, 724, 1153, 1820, 2891, 4572, 7249, 11482, 18190, 28821, 45651, 72338, 114582, 181549, 287596, 455647, 721847, 1143588, 1811748, 2870239, 4547232, 7203907, 11412882, 18080833, 28644680, 45380392, 71894054, 113898439, 180443915, 285869028, 452888824, 717490903, 1136687237

Offset: 0

Ilya Gutkovskiy, Dec 28 2016

nonn

Number of compositions (ordered partitions) into prime powers (1 excluded).

			a(6) = 4 because we have [4, 2], [3, 3], [2, 4] and [2, 2, 2].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for sequences related to compositions

Cf. A001221, A023360, A023894, A246655.

nmax = 48; CoefficientList[Series[1/(1 - Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

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

A002124 Number of compositions of n into a sum of odd primes.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 3, 7, 7, 8, 14, 15, 21, 28, 33, 47, 58, 76, 103, 125, 169, 220, 277, 373, 476, 616, 810, 1037, 1361, 1763, 2279, 2984, 3846, 5006, 6521, 8428, 10983, 14249, 18480, 24048, 31178, 40520, 52635, 68281, 88765, 115211, 149593, 194381, 252280, 327696, 425587, 552527, 717721

Offset: 0

N. J. A. Sloane

nonn

Arises in studying the Goldbach conjecture.

The g.f. -(z-1)*(z+1)*(z**2+z+1)*(z**2-z+1)/(1-z**6-z**3-z**5-z**7+z**9) conjectured by Simon Plouffe in his 1992 dissertation is wrong.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..1000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 300
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, II, pp. 354-382] [The sequence i_n]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to Goldbach conjecture

Cf. A002125, A023360, A024939, A077608.

Cf. A065091.

import Data.List (genericIndex)
a002124 n = genericIndex a002124_list n
a002124_list = 1 : f 1 [] a065091_list where
   f x qs ps'@(p:ps)
     | p <= x    = f x (p:qs) ps
     | otherwise = sum (map (a002124 . (x -)) qs) : f (x + 1) qs ps'
-- Reinhard Zumkeller, Mar 21 2014

A002124 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j),j=2..n)),z=0,n+1),z,n) end;
M:=120; a:=array(0..M); a[0]:=1; a[1]:=0; a[2]:=0; for n from 3 to M do t1:=0; for k from 2 to n do p := ithprime(k); if p <= n then t1 := t1 + a[n-p]; fi; od: a[n]:=t1; od: [seq(a[n],n=0..M)]; # N. J. A. Sloane, after MacMahon, Dec 03 2006; used in A002125

a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = (s = 0; p = 3; While[p <= n, s = s + a[n-p]; p = NextPrime[p]]; s); a /@ Range[0, 58] (* Jean-François Alcover, Jun 28 2011, after P. A. MacMahon *)

a(0)=1, a(1)=a(2)=0; for n >= 3, a(n) = Sum_{ primes p with 3 <= p <= n} a(n-p). [MacMahon]

G.f.: 1/( 1 - Sum_{k>=2} x^A000040(k) ). [Joerg Arndt, Sep 30 2012]

Better description and more terms from Philippe Flajolet, Nov 11 2002

Edited by N. J. A. Sloane, Dec 03 2006

A077608 Number of compositions of n into twin primes (i.e., primes that are members of a twin prime pair, like 3, 5, 7, 11, 13, but not 2 or 23).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 3, 7, 7, 8, 14, 15, 21, 28, 33, 47, 58, 75, 103, 125, 167, 220, 275, 370, 474, 610, 806, 1028, 1347, 1752, 2253, 2954, 3812, 4944, 6451, 8329, 10841, 14077, 18226, 23720, 30745, 39903, 51857, 67214, 87313, 113340, 147017, 190974

Offset: 0

Philippe Flajolet, Nov 11 2002

nonn

			a(15) = 8 since 15 = 11+7 = 7+11 = 5+13 = 13+5 = 3+5+7 = 3+7+5 = 5+3+7 = 5+7+3 = 7+3+5 = 7+5+3 and 3,5,7,11 belong to twin pairs.

Iain Fox, Table of n, a(n) for n = 0..8834 (first 1001 terms from Alois P. Heinz)
P. Flajolet, Publications

Cf. A001097, A002124, A023360.

A077608 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j)* subs([true=1,false=0],evalb(isprime(ithprime(j)-2) or isprime(ithprime(j)+2))),j=2..n+2)),z=0,n+1),z,n): end;

a[n_] := Coefficient[Series[ 1/(1 - Sum[z^Prime[j]*Boole[ PrimeQ[Prime[j] - 2] || PrimeQ[ Prime[j] + 2]], {j, 2, n + 2}]), {z, 0, n + 1}], z, n]; Table[a[n], {n, 0, 53}] (* Jean-François Alcover, Nov 09 2012, after Maple *)

ok(n)={isprime(n) && (isprime(n-2) || isprime(n+2))}
{my(n=60); Vec(1/(1-sum(k=1, n, if(ok(k), x^k, 0))) + O(x*x^n))} \\ Andrew Howroyd, Dec 28 2017

G.f.: 1/(1 - Sum_{k>=1} x^A001097(k)). - Andrew Howroyd, Dec 28 2017

A353401 Number of integer compositions of n with all prime run-lengths.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 4, 3, 6, 9, 10, 18, 27, 35, 54, 83, 107, 176, 242, 354, 515, 774, 1070, 1648, 2332, 3429, 4984, 7326, 10521, 15591, 22517, 32908, 48048, 70044, 101903, 149081, 216973, 316289, 461959, 672664, 981356, 1431256, 2086901, 3041577, 4439226, 6467735

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(0) = 1 through a(9) = 9 compositions (empty column indicated by dot, 0 is the empty composition):
  0   .  11   111   22   11111   33     11122     44       333
                                 222    22111     1133     11133
                                 1122   1111111   3311     33111
                                 2211             11222    111222
                                                  22211    222111
                                                  112211   1111122
                                                           1112211
                                                           1122111
                                                           2211111

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The case of runs equal to 2 is A003242 aerated.
The <= 1 version is A003242 ranked by A333489.
The version for parts instead of run-lengths is A023360, both A353429.
The version for partitions is A055923.
The > 1 version is A114901, ranked by A353427.
The <= 2 version is A128695, matching A335464.
The > 2 version is A353400, partitions A100405.
Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351013, A351017.
A005811 counts runs in binary expansion.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A167606 counts compositions with adjacent parts coprime.
A329738 counts uniform compositions, partitions A047966.
Cf. A078012, A165413, A175413, A274174, A333381, A333755, A353390, A353391, A353392, A353402, A353403.

b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h, add(
     `if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=1..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, May 18 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Length/@Split[#],_?(!PrimeQ[#]&)]&]],{n,0,15}]

a(21)-a(45) from Alois P. Heinz, May 18 2022

A347741 Number of compositions (ordered partitions) of n into at most 4 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 6, 6, 10, 15, 15, 24, 25, 31, 36, 35, 45, 50, 55, 63, 69, 73, 90, 91, 99, 113, 114, 130, 143, 144, 153, 166, 159, 195, 179, 213, 191, 253, 216, 279, 236, 288, 247, 343, 248, 377, 275, 397, 318, 456, 307, 503, 342, 524, 401, 572, 379, 641, 396, 667

Offset: 0

Ilya Gutkovskiy, Sep 11 2021

nonn

Cf. A000040, A023360, A121303, A340960, A347739, A347740, A347742, A347743.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,4,Prime@Range@PrimePi@n],1],{n,0,60}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

a(n) = Sum_{k=1..4} A121303(n,k) for n >= 2. - Alois P. Heinz, Sep 11 2021

A347742 Number of compositions (ordered partitions) of n into at most 5 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 6, 6, 10, 16, 20, 34, 40, 56, 72, 85, 110, 120, 145, 173, 194, 228, 260, 291, 340, 383, 414, 480, 518, 579, 653, 696, 759, 835, 875, 973, 1041, 1093, 1201, 1269, 1406, 1448, 1617, 1593, 1818, 1822, 2035, 1997, 2318, 2166, 2647, 2453, 2897, 2689, 3277

Offset: 0

Ilya Gutkovskiy, Sep 11 2021

nonn

Cf. A000040, A023360, A121303, A340961, A347739, A347740, A347741, A347743.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,5,Prime@Range@PrimePi@n],1],{n,0,60}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

a(n) = Sum_{k=1..5} A121303(n,k) for n >= 2. - Alois P. Heinz, Sep 11 2021

cp's OEIS Frontend

A265112 a(n) = A023360(A000040(n)): number of compositions of prime(n) into prime parts.

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A052284 Number of compositions of n into nonprime numbers.

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A280917 Expansion of 1/(1 - x - Sum_{k>=1} x^prime(k)).

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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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A280195 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(k))*x^k), where omega(k) is the number of distinct prime factors (A001221).

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A002124 Number of compositions of n into a sum of odd primes.

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A077608 Number of compositions of n into twin primes (i.e., primes that are members of a twin prime pair, like 3, 5, 7, 11, 13, but not 2 or 23).

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