A219180 -id:A219180 - cp's OEIS frontend

A000586 Number of partitions of n into distinct primes.

1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 5, 6, 7, 6, 9, 7, 9, 9, 9, 11, 11, 11, 13, 12, 14, 15, 15, 17, 16, 18, 19, 20, 21, 23, 22, 25, 26, 27, 30, 29, 32, 32, 35, 37, 39, 40, 42, 44, 45, 50, 50, 53, 55, 57, 61, 64, 67, 70, 71, 76, 78, 83, 87, 89, 93, 96

Offset: 0

N. J. A. Sloane

			n=16 has a(16) = 3 partitions into distinct prime parts: 16 = 2+3+11 = 3+13 = 5+11.

H. Gupta, Certain averages connected with partitions. Res. Bull. Panjab Univ. no. 124 1957 427-430.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in two entries, N0004 and N0039).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
H. Gupta, Partitions into distinct primes, Proc. Nat. Acad. Sci. India, 21 (1955), 185-187. [broken link]
BongJu Kim, Partition number identities which are true for all set of parts, arXiv:1803.08095 [math.CO], 2018.
Vaclav Kotesovec, Plot of log(a(n)) / log(Qas(n)) for n = 2..10^8, for Qas see the formula (25) from the article by Murthy, Brack and Bhaduri, p. 7.
M. V. N. Murthy, M. Brack, R. K. Bhaduri, On the asymptotic distinct prime partitions of integers, arXiv:1904.02776 [math.NT], Mar 22 2019.
K. F. Roth and G. Szekeres, Some asymptotic formulae in the theory of partitions, Quart. J. Math., Oxford Ser. (2) 5 (1954), 241-259.

Cf. A000041, A070215, A000607 (parts may repeat), A112022, A000009, A046675, A319264, A319267.

a000586 = p a000040_list where
   p _  0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Aug 05 2012

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(ithprime(i)>n, 0, b(n-ithprime(i), i-1))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 15 2012

CoefficientList[Series[Product[(1+x^Prime[k]), {k, 24}], {x, 0, Prime[24]}], x]
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + If[Prime[i] > n, 0, b[n - Prime[i], i-1]]]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
nmax = 100; pmax = PrimePi[nmax]; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; poly[[3]] = 1; Do[p = Prime[k]; Do[poly[[j]] += poly[[j - p]], {j, nmax + 1, p + 1, -1}];, {k, 2, pmax}]; poly (* Vaclav Kotesovec, Sep 20 2018 *)

a(n,k=n)=if(n<1, !n, my(s);forprime(p=2,k,s+=a(n-p,p-1));s) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import isprime, primerange
from functools import cache
@cache
def a(n, k=None):
    if k == None: k = n
    if n < 1: return int(n == 0)
    return sum(a(n-p, p-1) for p in primerange(1, k+1))
print([a(n) for n in range(83)]) # Michael S. Branicky, Sep 03 2021 after Charles R Greathouse IV

G.f.: Product_{k>=1} (1+x^prime(k)).
a(n) = A184171(n) + A184172(n). - R. J. Mathar, Jan 10 2011
a(n) = Sum_{k=0..A024936(n)} A219180(n,k). - Alois P. Heinz, Nov 13 2012
log(a(n)) ~ Pi*sqrt(2*n/(3*log(n))) [Roth and Szekeres, 1954]. - Vaclav Kotesovec, Sep 13 2018

Entry revised by N. J. A. Sloane, Jun 10 2012

A117278 Triangle read by rows: T(n,k) is the number of partitions of n into k prime parts (n>=2, 1<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 1, 1, 0, 2, 2, 1, 0, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 0, 2, 1, 3, 2, 1, 1, 0, 1, 3, 2, 3, 2, 1, 0, 2, 2, 3, 3, 2, 1, 1, 1, 0, 4, 3, 3, 3, 2, 1, 0, 2, 2, 4, 3, 4, 2, 1, 1, 1, 1, 3, 4, 5, 3, 3, 2, 1, 0, 2, 2, 6, 4, 4, 4, 2, 1, 1, 0, 1, 5, 3, 6

Offset: 2

Emeric Deutsch, Mar 07 2006

Row n has floor(n/2) terms. Row sums yield A000607. T(n,1) = A010051(n) (the characteristic function of the primes). T(n,2) = A061358(n). Sum(k*T(n,k), k>=1) = A084993(n).

			T(12,3) = 2 because we have [7,3,2] and [5,5,2].
Triangle starts:
  1;
  1;
  0, 1;
  1, 1;
  0, 1, 1;
  1, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1;
  ...

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

Columns k=1-10 give: A010051, A061358, A068307, A259194, A259195, A259196, A259197, A259198, A259200, A259201.
Row sums give A000607.
T(A000040(n),n) gives A259254(n).
Cf. A084993, A219180.

g:=1/product(1-t*x^(ithprime(j)),j=1..30): gser:=simplify(series(g,x=0,30)): for n from 2 to 22 do P[n]:=sort(coeff(gser,x^n)) od: for n from 2 to 22 do seq(coeff(P[n],t^j),j=1..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i))[]], 0)))
    end:
T:= n-> subsop(1=NULL, b(n, numtheory[pi](n)))[]:
seq(T(n), n=2..25);  # Alois P. Heinz, Nov 16 2012

(* As triangle: *) nn=20;a=Product[1/(1-y x^i),{i,Table[Prime[n],{n,1,nn}]}];Drop[CoefficientList[Series[a,{x,0,nn}],{x,y}],2,1]//Grid (* Geoffrey Critzer, Oct 30 2012 *)

parts(n, pred)={prod(k=1, n, if(pred(k), 1/(1-y*x^k) + O(x*x^n), 1))}
{my(n=15); apply(p->Vecrev(p/y), Vec(parts(n, isprime)-1))} \\ Andrew Howroyd, Dec 28 2017

G.f.: G(t,x) = -1+1/product(1-tx^(p(j)), j=1..infinity), where p(j) is the j-th prime.

A117929 Number of partitions of n into 2 distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 2, 0, 3, 1, 2, 0, 2, 0, 3, 1, 2, 1, 3, 0, 4, 0, 1, 1, 3, 0, 4, 1, 3, 1, 3, 0, 5, 1, 4, 0, 3, 0, 5, 1, 3, 0, 3, 0, 6, 1, 2, 1, 5, 0, 6, 0, 2, 1, 5, 0, 6, 1, 4, 1, 5, 0, 7, 0, 4, 1, 4, 0, 8, 1, 4, 0, 4, 0, 9, 1, 4, 0, 4, 0, 7, 0, 3, 1, 6, 0, 8, 1, 5, 1

Offset: 1

Emeric Deutsch, Apr 03 2006

Number of distinct rectangles with prime length and width such that L + W = n, W < L. For example, a(16) = 2; the two rectangles are 3 X 13 and 5 X 11. - Wesley Ivan Hurt, Oct 29 2017

			a(24) = 3 because we have [19,5], [17,7] and [13,11].

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A010051, A045917, A061358, A073610, A166081 (positions of 0), A077914 (positions of 2), A080862 (positions of 6).

Column k=2 of A219180. - Alois P. Heinz, Nov 13 2012

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j-1),j=1..35): gser:=series(g,x=0,130): seq(coeff(gser,x,n),n=1..125);
# alternative
A117929 := proc(n)
    local a,i,p ;
    a := 0 ;
    p := 2 ;
    for i from 1 do
        if 2*p >= n then
            return a;
        end if;
        if isprime(n-p) then
            a := a+1 ;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
seq(A117929(n),n=1..80) ; # R. J. Mathar, Oct 01 2021

l = {}; For[n = 1, n <= 1000, n++, c = 0; For[k = 1, Prime[k] < n/2, k++, If[PrimeQ[n - Prime[k]], c = c + 1] ]; AppendTo[l, c] ] l (* Jake Foster, Oct 27 2008 *)
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[#,PrimeQ]&&#[[1]]!= #[[2]] &)],{n,120}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 26 2020 *)

a(n)=my(s);forprime(p=2,(n-1)\2,s+=isprime(n-p));s \\ Charles R Greathouse IV, Feb 26 2014

from sympy import sieve
from collections import Counter
from itertools import combinations
def aupton(max):
    sieve.extend(max)
    a = Counter(c[0]+c[1] for c in combinations(sieve._list, 2))
    return [a[n] for n in range(1, max+1)]
print(aupton(105)) # Michael S. Branicky, Feb 16 2024

G.f.: Sum_{j>0} Sum_{i=1..j-1} x^(p(i)+p(j)), where p(k) is the k-th prime.
G.f.: A(x)^2/2 - A(x^2)/2 where A(x) = Sum_{p in primes} x^p. - Geoffrey Critzer, Nov 21 2012
a(n) = [x^n*y^2] Product_{i>=1} (1+x^prime(i)*y). - Alois P. Heinz, Nov 22 2012
a(n) = Sum_{i=2..floor((n-1)/2)} A010051(i) * A010051(n-i). - Wesley Ivan Hurt, Oct 29 2017

A219107 Number of compositions (ordered partitions) of n into distinct prime parts.

Original entry on oeis.org

1, 0, 1, 1, 0, 3, 0, 3, 2, 2, 8, 1, 8, 3, 8, 8, 10, 25, 16, 9, 16, 38, 16, 61, 18, 62, 46, 66, 160, 91, 138, 99, 70, 122, 306, 126, 314, 151, 362, 278, 588, 901, 602, 303, 654, 1142, 888, 1759, 892, 1226, 950, 2160, 1230, 3379, 1444, 2372, 2100, 4644, 7416

Offset: 0

Alois P. Heinz, Nov 11 2012

nonn

a(0) = 0 iff n in {1,4,6}.

			a(5) = 3: [2,3], [3,2], [5].
a(7) = 3: [2,5], [5,2], [7].
a(8) = 2: [3,5], [5,3].
a(9) = 2: [2,7], [7,2].
a(10) = 8: [2,3,5], [2,5,3], [3,2,5], [3,5,2], [5,2,3], [5,3,2], [3,7], [7,3].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000142, A000586, A032021, A218396.

with(numtheory):
b:= proc(n, i) b(n, i):=
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
a:= proc(n) local l; l:= b(n, pi(n));
      a(n):= add(l[i]*(i-1)!, i=1..nops(l))
    end:
seq(a(n), n=0..70);
# second Maple program:
s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 30 2020

zip = With[{m=Max[Length[#1], Length[#2]]}, PadRight[#1, m]+PadRight[#2, m] ]&;
b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, b[n, i-1] ~zip~ Join[{0}, If[Prime[i] > n, {}, b[n - Prime[i], i-1]]], {0}]];
a[n_] := Module[{l}, l = b[n, PrimePi[n]]; Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Mar 24 2017, adapted from Maple *)

a(n) = Sum_{k=0..A024936(n)} A219180(n,k)*k!.

A125688 Number of partitions of n into three distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 4, 3, 4, 2, 5, 3, 5, 4, 6, 1, 6, 3, 6, 4, 6, 3, 9, 3, 8, 5, 8, 4, 11, 3, 11, 5, 10, 3, 13, 3, 13, 6, 12, 2, 14, 5, 15, 6, 13, 2, 18, 5, 17, 6, 14, 4, 21, 5, 19, 7, 17, 4, 25, 4, 20, 8, 21, 4, 26, 4, 25, 9, 22, 4

Offset: 1

Reinhard Zumkeller, Nov 30 2006

a(A124868(n)) = 0; a(A124867(n)) > 0;

a(A125689(n)) = n and a(m) <> n for m < A125689(n).

			a(42) = #{2+3+37, 2+11+29, 2+17+23} = 3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

Column k=3 of A219180. - Alois P. Heinz, Nov 13 2012

Cf. A010051, A068307, A124868, A125689.

b:= proc(n, i) option remember; `if`(n=0, [1,0$3], `if`(i<1, [0$4],
       zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$3],
       b(n-ithprime(i), i-1)[1..3])[]], 0)))
    end:
a:= n-> b(n, numtheory[pi](n))[4]:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 15 2012

b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i<1, {0, 0, 0, 0}, Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, {0, 0, 0}, Take[b[n-Prime[i], i-1], 3]]]}]]]; a[n_] := b[n, PrimePi[n]][[4]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)
dp3Q[{a_,b_,c_}]:=Length[Union[{a,b,c}]]==3&&AllTrue[{a,b,c},PrimeQ]; Table[ Count[IntegerPartitions[n,{3}],?dp3Q],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Jan 30 2019 *)

a(n)=my(s);forprime(p=n\3,n-4,forprime(q=(n-p)\2+1,min(n-p,p-1),if(isprime(n-p-q),s++)));s \\ Charles R Greathouse IV, Aug 27 2012

From Alois P. Heinz, Nov 22 2012: (Start)

G.f.: Sum_{0

a(n) = [x^n*y^3] Product_{i>=1} (1+x^prime(i)*y). (End)

a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} A010051(i) * A010051(k) * A010051(n-i-k). - Wesley Ivan Hurt, Mar 29 2019

A184171 Number of partitions of n into an even number of distinct primes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 4, 4, 5, 5, 4, 6, 5, 5, 6, 7, 7, 8, 7, 9, 8, 9, 8, 11, 11, 12, 10, 13, 12, 14, 14, 15, 16, 17, 16, 20, 19, 20, 20, 24, 22, 26, 23, 27, 27, 30, 28, 34, 33, 36, 34, 40, 37, 43, 41, 46, 46, 50, 47, 56, 55

Offset: 0

Emeric Deutsch, (suggested by R. J. Mathar), Jan 09 2011

nonn

			a(33) = 5 because we have [31,2], [23,5,3,2], [19,7,5,2], [17,11,3,2], and [13,11,7,2].

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Alois P. Heinz)

Cf. A000586, A184172, A184198.

g := 1/2*(Product(1+z^ithprime(k), k = 1 .. 120)+Product(1-z^ithprime(k), k = 1 .. 120)): gser := series(g, z = 0, 110): seq(coeff(gser, z, n), n = 0 .. 85);
# second Maple program
with(numtheory):
b:= proc(n, i) option remember;
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
a:= proc(n) local l; l:= b(n, pi(n));
      add(l[2*i-1], i=1..iquo(nops(l)+1,2))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 15 2012

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, {}, b[n-Prime[i], i-1]]]}]]]; a[n_] := Module[{l}, l = b[n, PrimePi[n]]; Sum[l[[2*i-1]], {i, 1, Quotient[Length[l]+1, 2]}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

parts(n, pred, y)={prod(k=1, n, 1 + if(pred(k), y*x^k + O(x*x^n), 0))}
{my(n=80); Vec(parts(n, isprime, 1) + parts(n, isprime, -1))/2} \\ Andrew Howroyd, Dec 28 2017

G.f.: (1/2)*[Product_{k>=1} (1+z^prime(k)) + Product_{k>=1} (1-z^prime(k))].
a(n) = Sum_{k>=0} A219180(n,2*k). - Alois P. Heinz, Nov 15 2012
a(n) + A184172(n) = A000586(n). - R. J. Mathar, Mar 31 2023

A184172 Number of partitions of n into an odd number of distinct primes.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 4, 3, 5, 3, 4, 4, 5, 5, 6, 6, 7, 5, 7, 7, 8, 8, 8, 9, 11, 9, 10, 11, 12, 12, 14, 13, 16, 14, 16, 15, 19, 17, 20, 20, 22, 20, 23, 24, 27, 26, 28, 27, 33, 30, 34, 34, 37, 36, 41, 40, 46, 43, 47, 46, 55, 50, 56

Offset: 0

Emeric Deutsch (suggested by R. J. Mathar), Jan 09 2011

nonn

			a(33) = 4 because we have [23,7,3], [19,11,3], [17,13,3], and [17,11,5].

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Alois P. Heinz)

Cf. A000586, A184171, A184199.

g := 1/2*(Product(1+z^ithprime(k), k = 1 .. 120)-Product(1-z^ithprime(k), k = 1 .. 120)): gser := series(g, z = 0, 110): seq(coeff(gser, z, n), n = 0 .. 85);
# second Maple program
with(numtheory):
b:= proc(n, i) option remember;
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
a:= proc(n) local l; l:= b(n, pi(n));
      add(l[2*i], i=1..iquo(nops(l), 2))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 15 2012

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, {}, b[n-Prime[i], i-1]]]}]]]; a[n_] := Module[{l}, l = b[n, PrimePi[n]]; Sum[l[[2*i]], {i, 1, Quotient[Length[l], 2]}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

parts(n, pred, y)={prod(k=1, n, 1 + if(pred(k), y*x^k + O(x*x^n), 0))}
{my(n=80); (Vec(parts(n, isprime, 1)) - Vec(parts(n, isprime, -1)))/2} \\ Andrew Howroyd, Dec 28 2017

G.f.: (1/2)*[Product_{k>=1} (1+z^prime(k)) - Product_{k>=1} (1-z^prime(k))].
a(n) = Sum_{k>=0} A219180(n,2*k+1). - Alois P. Heinz, Nov 15 2012
a(n) + A184171(n) = A000586(n). - R. J. Mathar, Mar 31 2023

A219199 Number of partitions of n into 5 distinct primes.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 4, 0, 4, 1, 4, 2, 6, 1, 6, 2, 6, 4, 8, 2, 10, 5, 9, 6, 11, 5, 13, 6, 14, 10, 16, 9, 18, 11, 19, 15, 21, 14, 22, 16, 25, 22, 26, 20, 29, 25, 31, 29, 32, 29, 35, 34, 39, 39, 38, 39, 43, 45, 48, 50, 46, 53, 53, 57, 57, 66, 55

Offset: 28

Alois P. Heinz, Nov 14 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 28..10000

Column k=5 of A219180.

b:= proc(n, i) option remember; `if`(n=0, [1,0$5], `if`(i<1, [0$6],
       zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$5],
       b(n-ithprime(i), i-1)[1..5])[]], 0)))
    end:
a:= n-> b(n, numtheory[pi](n))[6]:
seq(a(n), n=28..100);

k = 5; b[n_, i_] := b[n, i] = If[n == 0, Join[{1}, Array[0&, k]], If[i<1, Array[0&, k+1], Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, Array[0&, k], Take[b[n-Prime[i], i-1], k]]]}]]]; a[n_] := b[n, PrimePi[n]][[k+1]]; Table[a[n], {n, 28, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)
Table[Length@Select[IntegerPartitions[k,{5}, Prime@Range@100], #[[1]] > #[[2]] > #[[3]] > #[[4]] > #[[5]] &], {k, 28, 100}] (* Robert Price, Apr 25 2025 *)

G.f.: Sum_{0

a(n) = [x^n*y^5] Product_{i>=1} (1+x^prime(i)*y).

A024936 a(n) = maximal length of partitions of n into distinct primes or -1 if there is no such partition.

Original entry on oeis.org

0, -1, 1, 1, -1, 2, -1, 2, 2, 2, 3, 1, 3, 2, 3, 3, 3, 4, 3, 3, 3, 4, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 4, 4, 5, 4, 5, 4, 5, 5, 5, 6, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 7, 6, 7, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 8, 7, 7, 7, 8, 7, 8, 7, 7, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 8, 8, 9, 8, 7, 8, 8, 8, 9, 8, 9, 8, 9, 8

Offset: 0

Clark Kimberling

sign

			a(12) = 3 because 12 = 2+3+7, but 12 is not a sum of 4 or more distinct primes.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

a(n) + 1 = row length of A219180(n).

ReplaceAll[Table[Max[Length /@ Select[IntegerPartitions[n, n, Prime[Range[n]]],  DuplicateFreeQ[#] &]], {n, 0, 100}] /. -Infinity -> -1] (* Robert Price, Apr 23 2025 *)

More terms from Naohiro Nomoto, Oct 28 2001
More terms from David Wasserman, Jan 23 2003
Offset changed and edited by Alois P. Heinz, Nov 13 2012

A219198 Number of partitions of n into 4 distinct primes.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 1, 3, 0, 3, 2, 4, 2, 4, 2, 5, 4, 5, 4, 6, 4, 6, 6, 6, 6, 6, 6, 9, 8, 8, 10, 8, 9, 11, 11, 11, 13, 10, 14, 13, 16, 13, 18, 12, 19, 14, 21, 15, 22, 13, 25, 18, 26, 17, 29, 14, 31, 21, 32, 19, 35, 17, 39, 25, 38, 20, 43, 21, 48, 26

Offset: 17

Alois P. Heinz, Nov 14 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 17..10000

Column k=4 of A219180.

b:= proc(n, i) option remember; `if`(n=0, [1,0$4], `if`(i<1, [0$5],
       zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$4],
       b(n-ithprime(i), i-1)[1..4])[]], 0)))
    end:
a:= n-> b(n, numtheory[pi](n))[5]:
seq(a(n), n=17..100);

k = 4; b[n_, i_] := b[n, i] = If[n == 0, Join[{1}, Array[0&, k]], If[i<1, Array[0&, k+1], Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, Array[0&, k], Take[b[n-Prime[i], i-1], k]]]}]]]; a[n_] := b[n, PrimePi[n]][[k+1]]; Table[a[n], {n, 17, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)
Table[Length@Select[IntegerPartitions[k,{4}, Prime@Range@100], #[[1]] > #[[2]] > #[[3]] > #[[4]] &], {k, 17, 100}] (* Robert Price, Apr 25 2025 *)

G.f.: Sum_{0

a(n) = [x^n*y^4] Product_{i>=1} (1+x^prime(i)*y).

Showing 1-10 of 24 results. Next

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A000586 Number of partitions of n into distinct primes.

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A117278 Triangle read by rows: T(n,k) is the number of partitions of n into k prime parts (n>=2, 1<=k<=floor(n/2)).

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A117929 Number of partitions of n into 2 distinct primes.

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A219107 Number of compositions (ordered partitions) of n into distinct prime parts.

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A125688 Number of partitions of n into three distinct primes.

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A184171 Number of partitions of n into an even number of distinct primes.

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