A302479 -id:A302479 - cp's OEIS frontend

A341461 Number of partitions of n into 3 distinct nonprime parts.

1, 0, 1, 1, 2, 1, 2, 2, 4, 3, 4, 4, 6, 5, 8, 7, 9, 9, 10, 12, 14, 14, 14, 17, 19, 19, 22, 23, 24, 28, 27, 31, 33, 35, 36, 40, 40, 44, 47, 49, 50, 55, 53, 61, 62, 66, 65, 73, 72, 79, 81, 86, 85, 95, 91, 101, 101, 109, 107, 119, 114, 125, 125, 134, 134

Offset: 11

Ilya Gutkovskiy, Feb 12 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 11..10000

Cf. A005171, A018252, A096258, A125688, A302479, A341408, A341462, A341464, 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, 3):
seq(a(n), n=11..75);  # 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, 3];
a /@ Range[11, 75] (* Jean-François Alcover, Jul 01 2021, after Alois P. Heinz *)

from functools import lru_cache
from sympy import isprime
@lru_cache(maxsize=None)
def b(n, i, t):
  if n == 0: return int(t == 0)
  if i < 1 or t < 1: return 0
  b2 = 0 if isprime(i) else b(n-i, min(n-i, i-1), t-1)
  return b(n, i-1, t) + b2
a = lambda n: b(n, n, 3)
print([a(n) for n in range(11, 76)]) # Michael S. Branicky, Feb 12 2021 after Alois P. Heinz

A341462 Number of partitions of n into 4 distinct nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 6, 10, 9, 13, 12, 17, 17, 21, 21, 28, 28, 34, 33, 42, 43, 51, 53, 61, 63, 73, 76, 87, 91, 102, 104, 119, 123, 137, 143, 157, 164, 179, 187, 205, 215, 232, 239, 262, 272, 294, 309, 327, 341, 365, 381, 406, 427, 448, 465

Offset: 19

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219198, A302479, A341451, A341461, A341464, 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, 4):
seq(a(n), n=19..78);  # 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, 4];
Table[a[n], {n, 19, 78}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n,{4}],Length[#]==Length[ Union[ #]] && NoneTrue[#,PrimeQ]&]],{n,19,80}] (* Harvey P. Dale, Nov 07 2021 *)

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 *)

A341465 Number of partitions of n into 6 distinct nonprime parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 6, 5, 8, 10, 13, 13, 18, 20, 26, 30, 36, 40, 49, 55, 65, 76, 88, 97, 114, 128, 146, 167, 187, 209, 237, 262, 294, 331, 366, 405, 449, 496, 547, 608, 663, 730, 798, 875, 953, 1050, 1136, 1239, 1342, 1463, 1577, 1723, 1849, 2008, 2159, 2334

Offset: 38

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219200, A302479, A341453, A341461, A341462, A341464, A341466, A341467.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i b(n$2, 6):
seq(a(n), n=38..95);  # 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, 6];
Table[a[n], {n, 38, 95}] (* Jean-François Alcover, Feb 23 2022, after Alois P. Heinz *)

A341466 Number of partitions of n into 7 distinct nonprime parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 6, 6, 9, 9, 14, 15, 21, 23, 30, 33, 43, 47, 61, 67, 81, 91, 112, 123, 150, 165, 194, 217, 255, 281, 330, 363, 417, 461, 529, 582, 665, 730, 823, 905, 1018, 1115, 1253, 1368, 1519, 1662, 1844, 2010, 2227, 2419, 2659, 2894, 3175, 3442

Offset: 50

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219201, A302479, A341454, A341461, A341462, A341464, A341465, 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, 7):
seq(a(n), n=50..105);  # 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, 7];
Table[a[n], {n, 50, 105}] (* Jean-François Alcover, Feb 23 2022, after Alois P. Heinz *)

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 *)

A358638 Number of partitions of n into at most 2 distinct nonprime parts.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 3, 1, 2, 2, 3, 3, 4, 2, 4, 3, 4, 4, 6, 3, 6, 5, 7, 5, 7, 5, 8, 6, 7, 7, 10, 7, 11, 7, 9, 9, 11, 8, 12, 9, 11, 10, 13, 9, 14, 11, 14, 11, 14, 11, 16, 13, 15, 13, 17, 13, 19, 14, 16, 15, 19, 15, 21, 15, 17, 17, 21, 16, 22, 17, 21, 18, 22, 18, 25, 18, 22

Offset: 0

Ilya Gutkovskiy, Nov 24 2022

nonn

Cf. A005171, A018252, A096258, A302479, A347788, A358639, A358640.

A358638(n) = if(n<2,1,!isprime(n)+sum(k=1,(n-1)\2,!(isprime(k)+isprime(n-k)))); \\ Antti Karttunen, Nov 25 2022

For n > 0, a(n) = A005171(n) + A302479(n).

A337853 a(n) is the number of partitions of n as the sum of two Niven numbers.

Original entry on oeis.org

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

Offset: 0

Marius A. Burtea, Sep 26 2020

a(n) >= 1 for n >= 2 ?.

For n <= 200000, a(n) = 1 only for n = 2, 3, 299, (2 = 1 + 1, 3 = 1 + 2, 299 = 1 + 288) and a(n) = 2 only for n in {4, 5, 35, 59, 79, 95, 97, 149, 169, 179, 389}.

			0 and 1 cannot be decomposed as the sum of two Niven numbers, so a(0) = a(1) = 0.
4 = 1 + 3 = 2 + 2 and 1, 2, 3 are in A005349, so a(4) = 2.
15 = 3 + 12 = 5 + 10 = 6 + 9 = 7 + 8 and 3, 5, 6, 7, 8, 9, 10, 12 are in A005349, so a(15) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A005349, A116357, A172151, A172398, A180149, A280829, A319468, A302479, A319395, A337854.

niven:=func; [#RestrictedPartitions(n,2,{k: k in [1..n-1] | niven(k)}): n in [0..100]];

m = 100; nivens = Select[Range[m], Divisible[#, Plus @@ IntegerDigits[#]] &]; a[n_] := Length[IntegerPartitions[n, {2}, nivens]]; Array[a, m, 0] (* Amiram Eldar, Sep 27 2020 *)

A341468 Number of partitions of n into 9 distinct nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 6, 7, 11, 12, 18, 20, 25, 30, 38, 45, 57, 67, 81, 95, 114, 133, 162, 187, 219, 255, 297, 343, 401, 462, 529, 607, 696, 793, 910, 1032, 1168, 1324, 1497, 1689, 1905, 2142, 2400, 2692, 3009, 3362, 3754, 4182, 4643, 5165

Offset: 79

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219203, A302479, A341457, A341461, A341462, A341464, A341465, A341466, A341467, A341469.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i b(n$2, 9):
seq(a(n), n=79..130);  # 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, 9];
Table[a[n], {n, 79, 130}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)

A341469 Number of partitions of n into 10 distinct nonprime parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 6, 7, 10, 11, 17, 17, 25, 28, 38, 44, 57, 64, 82, 95, 117, 136, 168, 189, 231, 264, 317, 366, 433, 490, 579, 660, 770, 877, 1019, 1146, 1327, 1497, 1720, 1940, 2215, 2481, 2825, 3165, 3583, 4008, 4523, 5033, 5664

Offset: 95

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219204, A302479, A341460, A341461, A341462, A341464, A341465, A341466, A341467, A341468.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i b(n$2, 10):
seq(a(n), n=95..145);  # Alois P. Heinz, Feb 12 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < t || 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, 10];
Table[a[n], {n, 95, 145}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A341461 Number of partitions of n into 3 distinct nonprime parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

A341462 Number of partitions of n into 4 distinct nonprime parts.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A341465 Number of partitions of n into 6 distinct nonprime parts.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A341466 Number of partitions of n into 7 distinct nonprime parts.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A358638 Number of partitions of n into at most 2 distinct nonprime parts.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A337853 a(n) is the number of partitions of n as the sum of two Niven numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A341468 Number of partitions of n into 9 distinct nonprime parts.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A341469 Number of partitions of n into 10 distinct nonprime parts.

Views