A005171 -id:A005171 - cp's OEIS frontend

A062602 Number of ways of writing n = p+c with p prime and c nonprime (1 or a composite number).

0, 0, 1, 1, 0, 2, 1, 2, 2, 1, 4, 3, 3, 3, 4, 2, 6, 3, 5, 4, 6, 3, 8, 3, 7, 4, 9, 5, 9, 4, 8, 7, 9, 4, 11, 3, 11, 9, 10, 6, 12, 5, 11, 8, 12, 7, 14, 5, 13, 7, 15, 9, 15, 6, 14, 10, 16, 9, 16, 5, 15, 13, 16, 8, 18, 6, 18, 15, 17, 9, 19, 8, 18, 12, 19, 11, 21, 7, 21, 14, 20, 13, 22, 7, 21, 14

Offset: 1

Labos Elemer, Jul 04 2001

			n = 22 has floor(n/2) = 11 partitions of form n = a + b; 3 partitions are of prime + prime [3 + 19 = 5 + 17 = 11 + 11], 3 partitions are of prime + nonprime [2 + 20 = 7 + 15 = 13 + 9], 5 partitions are nonprime + nonprime [1 + 21 = 4 + 18 = 6 + 16 = 8 + 14 = 10 + 12]. So a(22) = 3.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Goldbach conjecture

Cf. A061358, A014092, A062610, A224712.

Table[Length[Select[Range[Floor[n/2]], (PrimeQ[#] && Not[PrimeQ[n - #]]) || (Not[PrimeQ[#]] && PrimeQ[n - #]) &]], {n, 80}] (* Alonso del Arte, Apr 21 2013 *)
Table[Length[Select[IntegerPartitions[n,{2}],AnyTrue[#,PrimeQ] && !AllTrue[ #,PrimeQ]&]],{n,90}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 19 2020 *)

a(n+1) = SUM(A010051(k)*A005171(n-k+1): 1<=k<=n). [From Reinhard Zumkeller, Nov 05 2009]

a(n) + A061358(n) + A062610(n) = A004526(n). - R. J. Mathar, Sep 10 2021

A014684 In the sequence of positive integers subtract 1 from each prime number.

Original entry on oeis.org

1, 1, 2, 4, 4, 6, 6, 8, 9, 10, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 22, 22, 24, 25, 26, 27, 28, 28, 30, 30, 32, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 49, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 72

Offset: 1

Mohammad K. Azarian

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A048855, A113638.

a014684 n = n - fromIntegral (a010051 n)
-- Reinhard Zumkeller, Sep 10 2013

[n - (IsPrime(n) select 1 else 0): n in [1..80]]; // Bruno Berselli, Jul 18 2016

Table[If[PrimeQ[n],n-1,n],{n,100}] (* Harvey P. Dale, Aug 27 2015 *)

from sympy import isprime
def A014684(n): return n-int(isprime(n)) # Chai Wah Wu, Oct 14 2023

a(n) = A005171(n) + n - 1.
a(n) = phi(n!)/phi((n-1)!). - Vladeta Jovovic, Nov 30 2002
For n > 3: a(n) = A113523(n) = A179278(n). - Reinhard Zumkeller, Jul 08 2010
a(n) = n - A010051(n). - Reinhard Zumkeller, Sep 10 2013

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

A239968 a(n) = 0 unless n is a nonprime A018252(k) then a(n) = k.

Original entry on oeis.org

1, 0, 0, 2, 0, 3, 0, 4, 5, 6, 0, 7, 0, 8, 9, 10, 0, 11, 0, 12, 13, 14, 0, 15, 16, 17, 18, 19, 0, 20, 0, 21, 22, 23, 24, 25, 0, 26, 27, 28, 0, 29, 0, 30, 31, 32, 0, 33, 34, 35, 36, 37, 0, 38, 39, 40, 41, 42, 0, 43, 0, 44, 45, 46, 47, 48, 0, 49, 50, 51, 0, 52

Offset: 1

Reinhard Zumkeller, Mar 30 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005171, A010051, A018252, A057427, A066246, A049084, A026233, A062298.

import Data.List (unfoldr, genericIndex)
a239968 n = genericIndex a239968_list (n - 1)
a239968_list = unfoldr c (1, 1, a018252_list) where
   c (i, z, xs'@(x:xs)) | i == x = Just (z, (i + 1, z + 1, xs))
                        | i /= x = Just (0, (i + 1, z, xs'))

Module[{k = 0}, Array[If[!PrimeQ[#], ++k, 0] &, 100]] (* Paolo Xausa, Jul 31 2025 *)

a(n) = A066246(n) - A010051(n) + 1.
a(n) = A026233(n) - A049084(n);
A057427(a(n)) = A005171(n).
a(n) = A062298(n)*A005171(n). - Ridouane Oudra, Jul 29 2025

A302479 Number of partitions of n into two distinct nonprime parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 5, 3, 5, 4, 6, 4, 6, 5, 7, 6, 6, 6, 9, 6, 10, 7, 8, 8, 10, 8, 11, 9, 10, 9, 12, 9, 13, 10, 13, 10, 13, 11, 15, 12, 14, 12, 16, 13, 18, 14, 15, 14, 18, 14, 20, 15, 16, 16, 20, 16, 21, 17, 20, 17

Offset: 1

Wesley Ivan Hurt, Apr 08 2018

			a(16) = 3; 16 = 15+1 = 12+4 = 10+6, which are distinct nonprimes.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A005171, A010051, A062610, A358638.

Cf. also A341461, A341462, A341464, A341465, A341466, A341467.

Table[Sum[(1 - PrimePi[n - i] + PrimePi[n - i - 1]) (1 - PrimePi[i] + PrimePi[i - 1]), {i, Floor[(n - 1)/2]}], {n, 100}]
Table[Length[Select[IntegerPartitions[n,{2}],Length[Union[#]]==2&&Boole[PrimeQ[#]]=={0,0}&]],{n,80}] (* Harvey P. Dale, Dec 28 2023 *)

A302479(n) = sum(k=1,(n-1)\2,!(isprime(k)+isprime(n-k))); \\ Antti Karttunen, Nov 25 2022

a(n) = Sum_{i=1..floor((n-1)/2)} (1 - c(i)) * (1 - c(n-i)), where c = A010051.

For n > 0, a(n) = A358638(n) - A005171(n). - Antti Karttunen, Nov 25 2022

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

Original entry on oeis.org

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

A060476 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is nonprime.

Original entry on oeis.org

1, 8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 256, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 440, 456, 459, 472, 480

Offset: 1

N. J. A. Sloane, Sep 18 2008

nonn

The old entry with this sequence number was a duplicate of A005171.

The asymptotic density of this sequence is Sum_{c composite} (1/zeta(c) - 1/zeta(c-1)) = 0.1182437806... - Amiram Eldar, Oct 18 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A096432, A074661.

Cf. A010051, A051903, A055229.

a060476 n = a060476_list !! (n-1)
a060476_list = filter ((== 0) . a010051' . (+ 1) . a051903) [1..]
-- Reinhard Zumkeller, Nov 30 2015

Join[{1}, Select[Range[500], !PrimeQ[1+Max[FactorInteger[#][[All, 2]]]]&]] (* Jean-François Alcover, Aug 02 2018 *)

isA060476(n) = if(n<2,1,!isprime(vecmax(factor(n)[,2])+1))

From Reinhard Zumkeller, Nov 30 2015: (Start)
A010051(A051903(a(n)+1)) = 1.
a(A055229(n)) > 1 for n > 1. (End)

A076608 Number of nonprimes k < n such that also n-k is not a prime.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 2, 1, 2, 4, 2, 3, 4, 4, 4, 7, 4, 7, 6, 7, 6, 10, 6, 11, 8, 12, 8, 13, 10, 15, 12, 13, 12, 18, 12, 21, 14, 16, 16, 21, 16, 23, 18, 21, 18, 24, 18, 27, 20, 27, 20, 27, 22, 31, 24, 29, 24, 32, 26, 37, 28, 30, 28, 37, 28, 41, 30, 33, 32, 41, 32, 43, 34, 40, 34, 43, 34

Offset: 1

Reinhard Zumkeller, Oct 21 2002

nonn

a(n) is odd iff n is even and n/2 is not prime.

Convolution of A005171 with itself. - R. J. Mathar, Sep 10 2021

			5=1+4=2+3=3+2=4+1, hence a(5)=2;
6=1+5=2+4=3+3=4+2=5+1, hence a(6)=0.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A073610, A005171.

Table[With[{nn=m},Total[Table[If[NoneTrue[{n,nn-n},PrimeQ],1,0],{n,nn-1}]]],{m,80}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 15 2020 *)

A341451 Number of partitions of n into 4 nonprime parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 5, 7, 8, 9, 10, 13, 13, 17, 17, 22, 21, 27, 27, 34, 34, 41, 40, 51, 49, 62, 59, 71, 70, 86, 82, 101, 97, 117, 112, 135, 131, 155, 150, 180, 170, 202, 196, 228, 222, 259, 248, 291, 281, 324, 314, 361, 348, 404, 388, 445, 431

Offset: 4

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A002095, A005171, A018252, A062610, A259194, A341408, A341452, A341453, A341454, A341455, A341457.

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), t-1))))
    end:
a:= n-> b(n$2, 4):
seq(a(n), n=4..69);  # 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], t - 1]]]];
a[n_] := b[n, n, 4];
Table[a[n], {n, 4, 69}] (* Jean-François Alcover, Aug 19 2021, after Alois P. Heinz *)

A341452 Number of partitions of n into 5 nonprime parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 6, 7, 9, 9, 12, 14, 16, 18, 22, 24, 29, 31, 38, 40, 49, 50, 62, 65, 77, 81, 97, 98, 120, 122, 144, 149, 176, 178, 212, 214, 251, 255, 299, 304, 352, 355, 412, 417, 482, 485, 559, 564, 643, 650, 742, 745, 850, 856, 965

Offset: 5

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A002095, A005171, A018252, A062610, A259195, A341408, A341451, A341453, A341454, A341455, A341457.

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), t-1))))
    end:
a:= n-> b(n$2, 5):
seq(a(n), n=5..68);  # 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], t - 1]]]];
a[n_] := b[n, n, 5];
Table[a[n], {n, 5, 68}] (* Jean-François Alcover, Aug 19 2021, after Alois P. Heinz *)

A341453 Number of partitions of n into 6 nonprime parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 6, 7, 9, 10, 12, 15, 16, 20, 23, 27, 30, 36, 40, 48, 53, 62, 68, 81, 87, 105, 112, 130, 141, 166, 176, 208, 219, 256, 271, 314, 331, 385, 403, 468, 488, 561, 588, 674, 702, 804, 837, 952, 991, 1126, 1168, 1321, 1372

Offset: 6

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A002095, A005171, A018252, A062610, A259196, A341408, A341451, A341452, A341454, A341455, A341457.

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), t-1))))
    end:
a:= n-> b(n$2, 6):
seq(a(n), n=6..67);  # 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], t - 1], 0]]];
a[n_] := b[n, n, 6];
Table[a[n], {n, 6, 67}] (* Jean-François Alcover, Feb 23 2022, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n,{6}],?(NoneTrue[#,PrimeQ]&)],{n,6,70}] (* _Harvey P. Dale, Feb 21 2023 *)

cp's OEIS Frontend

A062602 Number of ways of writing n = p+c with p prime and c nonprime (1 or a composite number).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A014684 In the sequence of positive integers subtract 1 from each prime number.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Python

Formula

Extensions

A239968 a(n) = 0 unless n is a nonprime A018252(k) then a(n) = k.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A302479 Number of partitions of n into two distinct nonprime parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

A060476 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is nonprime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A076608 Number of nonprimes k < n such that also n-k is not a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A341451 Number of partitions of n into 4 nonprime parts.

Views

Author