A219202 -id:A219202 - cp's OEIS frontend

A219180 Number T(n,k) of partitions of n into k distinct prime parts; triangle T(n,k), n>=0, read by rows.

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 1, 1, 1, 0, 0, 2, 2, 0, 0, 1, 2, 1, 0, 0, 2, 2, 0, 1, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 2, 2, 0, 0, 2, 3, 1

Offset: 0

Alois P. Heinz, Nov 13 2012

T(n,k) is defined for all n>=0 and k>=0. The triangle contains only elements with 0 <= k <= A024936(n). T(n,k) = 0 for k > A024936(n). Three rows are empty because there are no partitions of n into distinct prime parts for n in {1,4,6}.

			T(0,0) = 1: [], the empty partition.
T(2,1) = 1: [2].
T(5,1) = 1: [5], T(5,2) = 1: [2,3].
T(16,2) = 2: [5,11], [3,13].
Triangle T(n,k) begins:
  1;
  ;
  0, 1;
  0, 1;
  ;
  0, 1, 1;
  ;
  0, 1, 1;
  0, 0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 1;
  0, 0, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..1000, flattened

Columns k=0-10 give: A000007, A010051, A117929, A125688, A219198, A219199, A219200, A219201, A219202, A219203, A219204.
Row lengths are 1 + A024936(n).
Row sums give: A000586.
Last elements of rows give: A219181.
Row maxima give: A219182.
Least n with T(n,k) > 0 is A007504(k).
Cf. A000040, A117278, A219107.

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:
T:= proc(n) local l; l:= b(n, numtheory[pi](n));
       while nops(l)>0 and l[-1]=0 do l:= subsop(-1=NULL, l) od; l[]
    end:
seq(T(n), n=0..50);

nn=20;a=Table[Prime[n],{n,1,nn}];CoefficientList[Series[Product[1+y x^a[[i]],{i,1,nn}],{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Nov 21 2012 *)
zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]]; b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, zip[Plus, b[n, i-1], Join[{0}, If[Prime[i] > n, {}, b[n-Prime[i], i-1]]], 0]]]; T[n_] := Module[{l}, l = b[n, PrimePi[n]]; While[Length[l]>0 && l[[-1]] == 0, l = ReplacePart[l, -1 -> Sequence[]]]; l]; Table[T[n], {n, 0, 50}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)

T(n)={ Vec(prod(k=1, n, 1 + isprime(k)*y*x^k + O(x*x^n))) }
{ my(t=T(20)); for(n=1, #t, print(if(t[n]!=0, Vecrev(t[n]), []))) } \\ Andrew Howroyd, Dec 22 2017

G.f. of column k: Sum_{0

T(n,k) = [x^n*y^k] Product_{i>=1} (1+x^prime(i)*y).

A341467 Number of partitions of n into 8 distinct nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 6, 7, 10, 12, 16, 19, 24, 28, 36, 41, 52, 60, 73, 85, 102, 116, 142, 161, 192, 217, 256, 287, 339, 382, 442, 496, 574, 639, 737, 821, 937, 1041, 1184, 1309, 1483, 1640, 1845, 2037, 2283, 2508, 2807, 3081, 3430, 3761, 4170, 4553, 5045

Offset: 64

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219202, A302479, A341455, A341461, A341462, A341464, A341465, A341466.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(isprime(i), 0, b(n-i, min(n-i, i-1), t-1))))
    end:
a:= n-> b(n$2, 8):
seq(a(n), n=64..118);  # Alois P. Heinz, Feb 12 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[PrimeQ[i], 0, b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 8];
Table[a[n], {n, 64, 118}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341979 Number of partitions of n into 8 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 4, 0, 3, 0, 3, 1, 7, 0, 6, 1, 6, 1, 11, 0, 11, 2, 11, 3, 19, 1, 18, 3, 18, 5, 30, 4, 28, 6, 30, 10, 45, 6, 40, 11, 46, 16, 63, 11, 60, 19, 69, 25, 88, 18, 86, 32, 97, 36, 121, 32, 123, 47, 131, 55, 164, 49, 164, 69, 181, 80

Offset: 59

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A036497, A219202, A341951, A341973, A341974, A341975, A341976, A341977, A341978, A341980, A341981.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i-1)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 9)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 8):
seq(a(n), n=59..130);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i - 1]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 9}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 8];
Table[a[n], {n, 59, 130}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

A344989 Smallest number whose number of partitions into n distinct primes is n, or zero if there are no such partitions.

Original entry on oeis.org

2, 16, 26, 33, 55, 59, 0, 0, 124, 159, 233, 227, 276, 0, 372, 480, 0, 0, 0, 752, 0, 920, 0, 1011, 0, 1211, 1425, 0, 0, 0, 0, 0, 2050, 2336, 2495, 0, 0, 0, 0, 3340, 0, 3712, 0, 0, 4303, 0, 0, 0, 0, 5195, 0, 5669, 0, 6163, 6673, 0, 0, 0, 7504, 0, 0, 8670, 0, 9304, 9623, 0, 0, 0, 10638, 10981, 0, 12062, 0

Offset: 1

Metin Sariyar, Jun 04 2021

From David A. Corneth, Aug 21 2025: (Start)
How to prove a 0? I used the heuristic:
a(n) = 0 if 2*n consecutive integers can be written in strictly more than n ways as a sum of n distinct primes and up to that point no positive integer has exactly n such ways.
What other rules where used? (End)

			a(2) = 16 because 16 is the smallest number whose number of partitions into 2 distinct primes is 2; 16 = 3+13 = 5+11.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios! 233

Cf. A364692 asks for the largest number with the same properties.

Cf. A000586, A077914, A117929, A125688, A219180, A219198, A219199, A219200, A219201, A219202, A219203, A219204.

a(12)-a(20) from Alois P. Heinz, Jun 04 2021

More terms from David A. Corneth, Aug 21 2025

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A219180 Number T(n,k) of partitions of n into k distinct prime parts; triangle T(n,k), n>=0, read by rows.

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A341467 Number of partitions of n into 8 distinct nonprime parts.

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A341979 Number of partitions of n into 8 distinct primes (counting 1 as a prime).

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A344989 Smallest number whose number of partitions into n distinct primes is n, or zero if there are no such partitions.

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