A062610 -id:A062610 - cp's OEIS frontend

A061358 Number of ways of writing n = p+q with p, q primes and p >= q.

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, 2, 1, 2, 1, 3, 0, 3, 1, 3, 0, 2, 0, 3, 1, 2, 1, 4, 0, 4, 0, 2, 1, 3, 0, 4, 1, 3, 1, 4, 0, 5, 1, 4, 0, 3, 0, 5, 1, 3, 0, 4, 0, 6, 1, 3, 1, 5, 0, 6, 0, 2, 1, 5, 0, 6, 1, 5, 1, 5, 0, 7, 0, 4, 1, 5, 0, 8, 1, 5, 0, 4, 0, 9, 1, 4, 0, 5, 0, 7, 0, 3, 1, 6, 0, 8, 1, 5, 1

Offset: 0

Amarnath Murthy, Apr 28 2001

For an odd number n, a(n) = 0 if n-2 is not a prime, otherwise a(n) = 1.
For n > 1, a(2n) is at least 1, according to Goldbach's conjecture.
a(A014092(n)) = 0; a(A014091(n)) > 0; a(A067187(n)) = 1. - Reinhard Zumkeller, Nov 22 2004
Number of partitions of n into two primes.
Number of unordered ways of writing n as the sum of two primes.
a(2*n) = A068307(2*n+2). - Reinhard Zumkeller, Aug 08 2009
4*a(n) is the total number of divisors of all primes p and q such that n = p+q and p >= q. - Wesley Ivan Hurt, Mar 05 2016
Indices where a(n) = 0 correspond to A164376 UNION A025584. - Bill McEachen, Jan 31 2024

			a(22) = 3 because 22 can be written as 3+19, 5+17 and 11+11.

a(2n) is A045917.
Cf. A067187, A067188, A067189, A067190, A067191, A063610, A073610, A107318.
Column k=2 of A117278.

[#RestrictedPartitions(n,2,{p:p in PrimesUpTo(1000)}):n in [0..100] ] // Marius A. Burtea, Jan 19 2019

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j),j=1..30): gser:=series(g,x=0,110): seq(coeff(gser,x,n),n=0..105); # Emeric Deutsch, Apr 03 2006

a[n_] := Length[Select[n - Prime[Range[PrimePi[n/2]]], PrimeQ]]; Table[a[n], {n, 0, 100}] (* Paul Abbott, Jan 11 2005 *)
With[{nn=110},CoefficientList[Series[Sum[x^(Prime[i]+Prime[j]),{j,nn},{i,j}],{x,0,nn}],x]] (* Harvey P. Dale, Aug 17 2017 *)
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[#,PrimeQ]&)],{n,0,110}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 03 2021 *)

a(n)=my(s);forprime(q=2,n\2,s+=isprime(n-q));s \\ Charles R Greathouse IV, Mar 21 2013

from sympy import primerange, isprime, floor
def a(n):
    s=0
    for q in primerange(2, n//2 + 1): s+=isprime(n - q)
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 30 2017

G.f.: Sum_{j>0} Sum_{i=1..j} x^(p(i)+p(j)), where p(k) is the k-th prime. - Emeric Deutsch, Apr 03 2006
A065577(n) = a(10^n).
From Wesley Ivan Hurt, Jan 04 2013: (Start)
a(n) = Sum_{i=1..floor(n/2)} A010051(i) * A010051(n-i).
a(n) = Sum_{i=1..floor(n/2)} floor((A010051(i) + A010051(n-i))/2). (End)
a(n) + A062610(n) + A062602(n) = A004526(n). - R. J. Mathar, Sep 10 2021
a(n) = Sum_{k=floor((n-1)^2/4)+1..floor(n^2/4)} c(A339399(2k-1)) * c(A339399(2k)), where c = A010051. - Wesley Ivan Hurt, Jan 19 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001

Comments edited by Zak Seidov, May 28 2014

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

Original entry on oeis.org

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

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

A006307 Number of ways writing 2^n as unordered sums of 2 primes.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 3, 8, 11, 22, 25, 53, 76, 151, 244, 435, 749, 1314, 2367, 4239, 7471, 13705, 24928, 45746, 83467, 153850, 283746, 525236, 975685, 1817111, 3390038, 6341424, 11891654, 22336060, 42034097, 79287664, 149711134, 283277225, 536710100, 1018369893

Offset: 0

N. J. A. Sloane

nonn

			n = 5: 2^5 = 32 = 3+29 = 13+19 so a(5) = 2.

Bohman, Jan and Froberg, Carl-Erik; Numerical results on the Goldbach conjecture. Nordisk Tidskr. Informationsbehandling (BIT) 15 (1975), no. 3, 239-243.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Index entries for sequences related to Goldbach conjecture

Cf. A061358, A062602, A062610.

a:=proc(n) local c,k; c:=0: for k from 1 to floor((n-1)/2) do if isprime(2*k+1)=true and isprime(2*n-2*k-1)=true then c:=c+1 else c:=c fi od end: 0,0,1,seq(a(2*2^n),n=1..15); # Emeric Deutsch, Sep 22 2004

a(n)=my(N=2^n,s); forprime(q=2, N\2, s+=isprime(N-q)); s \\ Charles R Greathouse IV, Mar 02 2015

a(n) = A061358(2^n).

More terms from David W. Wilson
a(28)-a(35) from Ray Chandler, Feb 21 2004
a(36)=79287664 and a(37)=149711134 from Ray Chandler, Apr 10 2005
a(38)-a(40) from Russ Cox, Nov 04 2006

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 *)

A341457 Number of partitions of n into 9 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, 17, 20, 24, 28, 32, 38, 44, 51, 60, 67, 79, 91, 104, 120, 138, 154, 180, 203, 232, 262, 300, 335, 385, 428, 489, 543, 620, 688, 782, 861, 979, 1078, 1222, 1341, 1518, 1661, 1875, 2048, 2308

Offset: 9

Ilya Gutkovskiy, Feb 12 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..10000

Cf. A002095, A005171, A018252, A062610, A259200, A341408, A341451, A341452, A341453, A341454, A341455.

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, 9):
seq(a(n), n=9..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], 0]]];
a[n_] := b[n, n, 9];
Table[a[n], {n, 9, 68}] (* Jean-François Alcover, Feb 23 2022, after Alois P. Heinz *)

A341408 Number of partitions of n into 3 nonprime parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 2, 4, 5, 5, 5, 7, 6, 9, 8, 11, 10, 13, 12, 16, 14, 19, 16, 22, 19, 26, 22, 29, 27, 33, 28, 39, 33, 42, 38, 47, 43, 53, 45, 58, 52, 63, 59, 70, 61, 77, 68, 83, 76, 91, 79, 98, 88, 105, 95, 115, 102, 121, 111, 130, 119, 141, 124, 148

Offset: 3

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A002095, A005171, A018252, A062610, A068307, A341451, 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, 3):
seq(a(n), n=3..72);  # 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, 3];
a /@ Range[3, 72] (* Jean-François Alcover, Mar 28 2021, after Alois P. Heinz *)

A341454 Number of partitions of n into 7 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, 17, 20, 24, 27, 32, 37, 43, 49, 58, 64, 76, 85, 99, 111, 129, 140, 166, 182, 210, 230, 267, 290, 336, 362, 418, 451, 519, 559, 640, 685, 784, 837, 956, 1020, 1158, 1232, 1397, 1483, 1677, 1776

Offset: 7

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A002095, A005171, A018252, A062610, A259197, A341408, A341451, A341452, A341453, 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, 7):
seq(a(n), n=7..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, 7];
Table[a[n], {n, 7, 67}] (* Jean-François Alcover, Feb 23 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A061358 Number of ways of writing n = p+q with p, q primes and p >= q.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

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

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A006307 Number of ways writing 2^n as unordered sums of 2 primes.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A341451 Number of partitions of n into 4 nonprime parts.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A341452 Number of partitions of n into 5 nonprime parts.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A341453 Number of partitions of n into 6 nonprime parts.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A341457 Number of partitions of n into 9 nonprime parts.

Views

Author

Keywords

Links