A219199 -id:A219199 - 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).

A341464 Number of partitions of n into 5 distinct nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 5, 7, 7, 9, 12, 14, 15, 19, 21, 27, 29, 35, 38, 47, 49, 59, 65, 77, 82, 96, 102, 119, 128, 147, 157, 181, 189, 216, 231, 260, 276, 309, 327, 366, 387, 431, 454, 505, 529, 584, 617, 678, 713, 780, 818, 892, 938, 1020, 1071, 1164, 1213, 1311, 1378

Offset: 28

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219199, A302479, A341452, A341461, A341462, A341465, A341466, A341467.

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, 5):
seq(a(n), n=28..88);  # 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]]]];
a[n_] := b[n, n, 5];
Table[a[n], {n, 28, 88}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)

A341976 Number of partitions of n into 5 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 2, 1, 3, 1, 4, 0, 3, 2, 6, 2, 6, 2, 7, 5, 9, 4, 10, 5, 10, 8, 12, 7, 12, 8, 15, 12, 16, 12, 18, 14, 20, 17, 22, 18, 23, 20, 27, 26, 29, 27, 30, 30, 33, 36, 36, 36, 35, 41, 43, 48, 43, 49, 43, 56, 52, 61, 51, 64, 52, 73, 64, 77, 58, 82, 64, 93

Offset: 18

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A036497, A219199, A341948, A341973, A341974, A341975, A341977.

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, 6)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 5):
seq(a(n), n=18..91);  # 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, 6}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 5];
Table[a[n], {n, 18, 100}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)

A358010 Number of partitions of n into at most 5 distinct prime parts.

Original entry on oeis.org

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, 13, 15, 15, 17, 15, 18, 17, 20, 20, 23, 20, 25, 22, 27, 28, 28, 27, 30, 29, 36, 34, 38, 36, 41, 35, 48, 41, 48, 44, 50, 46, 58, 53, 61, 54, 64, 55, 72, 66, 74

Offset: 0

Ilya Gutkovskiy, Oct 24 2022

nonn

Cf. A000040, A000586, A219180, A219199, A223893, A347550, A347587, A347609, A347742, A358009, A358011.

A340001 Number of ways prime(n) is a sum of five distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 5, 6, 11, 14, 16, 25, 29, 39, 57, 68, 75, 88, 92, 109, 169, 198, 235, 240, 322, 331, 379, 437, 497, 565, 635, 634, 803, 798, 896, 888, 1091, 1328, 1477, 1444, 1616, 1753, 1730, 2080, 2262, 2452, 2627, 2588, 2790, 3043, 3004, 3535

Offset: 1

Michel Lagneau, Dec 26 2020

nonn

Conjecture: all primes >= 43 are the sum of five distinct primes.
The sequence of the prime numbers that are the sum of five distinct prime numbers begins with 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, ...
The primes in the sequence are 2, 5, 11, 29, 109, 331, 379, 1091, 1753, ...
The squares in the sequence are 0, 1, 16, 25, 169, 1444, ...

			a(14) = 1 because prime(14) = 43 = 3 + 5 + 7 + 11 + 17.
a(17) = 5 because prime(17) = 59 = 3 + 5 + 7 + 13 + 31 = 3 + 5 + 11 + 17 + 23 = 3 + 7 + 13 + 17 + 19 = 5 + 7 + 11 + 13 + 23 = 5 + 7 + 11 + 17 + 19.

Alois P. Heinz, Table of n, a(n) for n = 1..3000
Wikipedia, Goldbach's weak conjecture

Cf. A000040, A002372, A002375, A024684, A062301, A178041, A219199, A224534.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+(p-> `if`(p>n, 0,
       x*b(n-p, i-1)))(ithprime(i)))), x, 6)
    end:
a:= n-> coeff(b(ithprime(n), n), x, 5):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 1, 0, b[n, i - 1] + Function[p, If[p > n, 0,
     x*b[n - p, i - 1]]][Prime[i]]]], {x, 0, 6}];
a[n_] := SeriesCoefficient[b[Prime[n], n], {x, 0, 5}];
Array[a, 100] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[p,{5}],AllTrue[#,PrimeQ]&&Length[Union[#]]==5&]],{p,Prime[Range[70]]}] (* Harvey P. Dale, Jul 07 2024 *)

a(n) = A219199(A000040(n)).

a(n) = [x^prime(n)*y^5] Product_{i>=1} (1+x^prime(i)*y). - Alois P. Heinz, Dec 30 2020

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

Showing 1-6 of 6 results.

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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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A341464 Number of partitions of n into 5 distinct nonprime parts.

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A341976 Number of partitions of n into 5 distinct primes (counting 1 as a prime).

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A358010 Number of partitions of n into at most 5 distinct prime parts.

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A340001 Number of ways prime(n) is a sum of five distinct primes.

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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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