A121303 -id:A121303 - cp's OEIS frontend

A340960 Number of ways to write n as an ordered sum of 4 primes.

1, 4, 6, 8, 13, 16, 22, 24, 22, 32, 34, 40, 47, 48, 56, 68, 70, 76, 90, 84, 111, 112, 126, 120, 144, 120, 176, 140, 184, 148, 226, 168, 264, 184, 262, 196, 313, 192, 352, 208, 366, 256, 418, 240, 473, 260, 496, 324, 536, 300, 616, 308, 634, 348, 670, 348, 772, 364, 786, 412

Offset: 8

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Column k=4 of A121303.

Cf. A000040, A010051, A073610, A098238, A259194, A340961, A340962, A340963, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 4):
seq(a(n), n=8..67);  # Alois P. Heinz, Jan 31 2021

nmax = 67; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (Sum_{k>=1} x^prime(k))^4.

A340961 Number of ways to write n as an ordered sum of 5 primes.

Original entry on oeis.org

1, 5, 10, 15, 25, 36, 50, 65, 70, 90, 110, 125, 155, 170, 200, 241, 270, 300, 350, 375, 435, 500, 530, 600, 640, 696, 760, 850, 840, 985, 990, 1170, 1160, 1370, 1250, 1570, 1445, 1760, 1600, 2000, 1710, 2340, 1950, 2555, 2165, 2876, 2320, 3340, 2560, 3595, 2880, 3985, 3050

Offset: 10

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 10..10000

Column k=5 of A121303.

Cf. A000040, A010051, A073610, A098238, A259195, A340960, A340962, A340963, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 5):
seq(a(n), n=10..62);  # Alois P. Heinz, Jan 31 2021

nmax = 62; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (Sum_{k>=1} x^prime(k))^5.

A340962 Number of ways to write n as an ordered sum of 6 primes.

Original entry on oeis.org

1, 6, 15, 26, 45, 72, 106, 150, 186, 236, 306, 366, 455, 540, 636, 782, 912, 1056, 1236, 1410, 1617, 1896, 2106, 2400, 2696, 2976, 3348, 3716, 4026, 4446, 4917, 5340, 5982, 6380, 7017, 7476, 8377, 8640, 9765, 9936, 11202, 11496, 13132, 12930, 15117, 14672, 17178, 16800, 19696

Offset: 12

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 12..10000

Column k=6 of A121303.

Cf. A000040, A010051, A073610, A098238, A259196, A340960, A340961, A340963, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 6):
seq(a(n), n=12..60);  # Alois P. Heinz, Jan 31 2021

nmax = 60; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 12] &

G.f.: (Sum_{k>=1} x^prime(k))^6.

A340963 Number of ways to write n as an ordered sum of 7 primes.

Original entry on oeis.org

1, 7, 21, 42, 77, 133, 210, 316, 434, 574, 770, 980, 1239, 1547, 1876, 2331, 2828, 3367, 4032, 4746, 5565, 6574, 7602, 8757, 10136, 11480, 13132, 14882, 16646, 18662, 20951, 23268, 26082, 28861, 31787, 35218, 38745, 42532, 46403, 50883, 54810, 60613, 65016, 71302, 76069

Offset: 14

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 14..10000

Column k=7 of A121303.

Cf. A000040, A010051, A073610, A098238, A259197, A340960, A340961, A340962, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 7):
seq(a(n), n=14..58);  # Alois P. Heinz, Jan 31 2021

nmax = 58; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 14] &

G.f.: (Sum_{k>=1} x^prime(k))^7.

A340964 Number of ways to write n as an ordered sum of 8 primes.

Original entry on oeis.org

1, 8, 28, 64, 126, 232, 392, 624, 925, 1296, 1800, 2416, 3158, 4088, 5152, 6504, 8142, 9976, 12216, 14784, 17738, 21296, 25272, 29736, 35023, 40768, 47328, 54832, 62728, 71744, 81796, 92736, 105078, 118664, 132924, 149424, 167002, 186144, 206852, 229272, 253023

Offset: 16

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 16..10000

Column k=8 of A121303.

Cf. A000040, A010051, A073610, A098238, A259198, A340960, A340961, A340962, A340963, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 8):
seq(a(n), n=16..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^8, {x, 0, nmax}], x] // Drop[#, 16] &

G.f.: (Sum_{k>=1} x^prime(k))^8.

A340965 Number of ways to write n as an ordered sum of 9 primes.

Original entry on oeis.org

1, 9, 36, 93, 198, 387, 696, 1170, 1845, 2740, 3960, 5562, 7566, 10125, 13248, 17133, 22014, 27774, 34776, 43173, 53010, 64869, 78696, 94617, 113415, 134946, 159552, 188164, 219960, 256041, 297180, 342846, 394614, 452595, 516276, 587997, 667938, 755109, 852444

Offset: 18

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 18..10000

Column k=9 of A121303.

Cf. A000040, A010051, A073610, A098238, A259200, A340960, A340961, A340962, A340963, A340964, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 9):
seq(a(n), n=18..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^9, {x, 0, nmax}], x] // Drop[#, 18] &

G.f.: (Sum_{k>=1} x^prime(k))^9.

A340966 Number of ways to write n as an ordered sum of 10 primes.

Original entry on oeis.org

1, 10, 45, 130, 300, 622, 1185, 2100, 3495, 5480, 8266, 12100, 17140, 23730, 32155, 42802, 56400, 73180, 93820, 119250, 149872, 187090, 231765, 284490, 347335, 421332, 507580, 608840, 725500, 859450, 1014473, 1190700, 1392100, 1621710, 1879950, 2172610, 2503580

Offset: 20

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 20..10000

Column k=10 of A121303.

Cf. A000040, A010051, A073610, A098238, A259201, A340960, A340961, A340962, A340963, A340964, A340965.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 10):
seq(a(n), n=20..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^10, {x, 0, nmax}], x] // Drop[#, 20] &

G.f.: (Sum_{k>=1} x^prime(k))^10.

A224344 Number T(n,k) of compositions of n using exactly k primes (counted with multiplicity); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 2, 5, 1, 3, 8, 5, 5, 13, 13, 1, 7, 23, 27, 7, 11, 39, 52, 25, 1, 17, 65, 99, 66, 9, 27, 106, 186, 151, 41, 1, 40, 177, 340, 323, 133, 11, 61, 293, 608, 666, 358, 61, 1, 92, 482, 1076, 1330, 867, 236, 13, 142, 781, 1894, 2581, 1971, 737, 85, 1

Offset: 0

Alois P. Heinz, May 23 2013

			A(5,1) = 8: [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2], [3,1,1], [1,3,1], [1,1,3], [5].
Triangle T(n,k) begins:
   1;
   1;
   1,   1;
   1,   3;
   2,   5,   1;
   3,   8,   5;
   5,  13,  13,   1;
   7,  23,  27,   7;
  11,  39,  52,  25,   1;
  17,  65,  99,  66,   9;
  27, 106, 186, 151,  41,  1;
  40, 177, 340, 323, 133, 11;
  ...

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

Column k=0 gives: A052284.
Row sums are: A011782.
Row lengths are: A008619.
T(floor(n/2)) = A093178(n).
T(2n,n-1) = A001844(n-1) for n>0.
Cf. A000040, A000079, A004526, A102291, A121303, A222656.

T:= proc(n) option remember; local j; if n=0 then 1
      else []; for j to n do zip((x, y)->x+y, %,
      [`if`(isprime(j), 0, NULL), T(n-j)], 0) od; %[] fi
    end:
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]]]]; T[n_] := T[n] =  Module[{j, pc}, If[n == 0, {1}, pc = {}; For[j = 1, j <= n, j++, pc = zip[Plus, pc, Join[If[PrimeQ[j], {0}, {}], T[n-j]], 0]]; pc]]; 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) = A102291(n).

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

A340960 Number of ways to write n as an ordered sum of 4 primes.

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A340961 Number of ways to write n as an ordered sum of 5 primes.

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A340962 Number of ways to write n as an ordered sum of 6 primes.

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A340963 Number of ways to write n as an ordered sum of 7 primes.

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A340964 Number of ways to write n as an ordered sum of 8 primes.

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A340965 Number of ways to write n as an ordered sum of 9 primes.

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A340966 Number of ways to write n as an ordered sum of 10 primes.

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A224344 Number T(n,k) of compositions of n using exactly k primes (counted with multiplicity); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

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