A061358 -id:A061358 - cp's OEIS frontend

A067187 Numbers that can be expressed as the sum of two primes in exactly one way.

4, 5, 6, 7, 8, 9, 12, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259, 265, 271

Offset: 1

Amarnath Murthy, Jan 10 2002

nonn

All primes + 2 are terms of this sequence. Is 12 the last even term? - Frank Ellermann, Jan 17 2002
A048974, A052147, A067187 and A088685 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003
Values of n such that A061358(n)=1. - Emeric Deutsch, Apr 03 2006

			4 is a term as 4 = 2+2, 15 is a term as 15 = 13+2.

Robert Price, Table of n, a(n) for n = 1..5002
Index entries for sequences related to Goldbach conjecture

Cf. A023036, A045917, A061358.
Subsequence of A014091.
Numbers that can be expressed as the sum of two primes in k ways for k=0..10: A014092 (k=0), this sequence (k=1), A067188 (k=2), A067189 (k=3), A067190 (k=4), A067191 (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j),j=1..80): gser:=series(g,x=0,280): a:=proc(n) if coeff(gser,x^n)=1 then n else fi end: seq(a(n),n=1..272); # Emeric Deutsch, Apr 03 2006

cQ[n_]:=Module[{c=0},Do[If[PrimeQ[n-i]&&PrimeQ[i],c++],{i,2,n/2}]; c==1]; Select[Range[4,271],cQ[#]&] (* Jayanta Basu, May 22 2013 *)
y = Select[Flatten@Table[Prime[i] + Prime[j], {i, 60}, {j, 1, i}], # < Prime[60] &]; Select[Union[y], Count[y, #] == 1 &] (* Robert Price, Apr 21 2025 *)

Edited by Frank Ellermann, Jan 17 2002

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

A100949 Number of partitions of n into a prime and a semiprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 3, 2, 2, 1, 2, 2, 5, 1, 2, 2, 3, 2, 4, 2, 3, 3, 5, 5, 4, 1, 2, 4, 5, 2, 4, 3, 5, 6, 4, 5, 6, 3, 4, 5, 6, 5, 4, 3, 4, 4, 8, 7, 6, 4, 3, 7, 8, 6, 4, 4, 3, 10, 7, 6, 7, 4, 6, 10, 7, 6, 5, 6, 4, 7, 8, 9, 7, 5, 6, 9, 8, 9, 4, 5, 7, 8, 9, 11, 8, 4, 4, 11, 12, 10, 6, 10, 7, 13, 9, 9, 6

Offset: 1

Reinhard Zumkeller, Nov 23 2004

Marnell conjectures that a(n) > 0 for n > 10 after analyzing "many thousands of whole numbers". I find no exceptions below 100 million. - Charles R Greathouse IV, May 04 2010

			a(21) = #{7+2*7, 11+2*5, 17+2*2} = 3.

Geoffrey R. Marnell, "Ten Prime Conjectures", Journal of Recreational Mathematics 33:3 (2004-2005), pp. 193-196.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A061358, A000040, A001358.

Cf. A010051.

a100949 n = sum $ map (a010051 . (n -)) $ takeWhile (< n) a001358_list
-- Reinhard Zumkeller, Jun 26 2013

Table[Count[Sort/@(PrimeOmega/@IntegerPartitions[n,{2}]),{1,2}],{n,110}] (* Harvey P. Dale, Mar 25 2018 *)

list(lim)=my(p=primes(primepi(lim)),sp=select(n->bigomega(n)==2, vector(lim\1,i,i)),x=O('x^(lim\1+1))+'x); concat([0,0,0,0,0], Vec(sum(i=1,#p,x^p[i])*sum(i=1,#sp,x^sp[i]))) \\ Charles R Greathouse IV, Jun 14 2013

A100951(n) <= A100950(n) <= a(n) <= min(A000720(n), A072000(n)).

a(n) = Sum_{i=1..floor(n/2)} A010051(i) * A064911(n-i) + A010051(n-i) * A064911(i). - Wesley Ivan Hurt, May 02 2019

A341945 Number of partitions of n into 2 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 0, 2, 1, 3, 1, 2, 0, 3, 1, 3, 1, 3, 0, 4, 1, 3, 0, 2, 0, 4, 1, 3, 1, 4, 0, 4, 0, 3, 1, 3, 0, 5, 1, 4, 1, 4, 0, 6, 1, 4, 0, 3, 0, 6, 1, 3, 0, 4, 0, 7, 1, 4, 1, 5, 0, 6, 0, 3, 1, 5, 0, 7, 1, 6, 1, 5, 0, 7, 0, 5, 1, 5, 0, 9, 1, 5, 0, 4, 0, 10

Offset: 2

Ilya Gutkovskiy, Feb 24 2021

Number of partitions of n into 2 noncomposite numbers, A008578. - Antti Karttunen, Dec 13 2021

Antti Karttunen, Table of n, a(n) for n = 2..20000

Cf. A001031, A008578, A034891, A061358, A341946, A341947, A341948, A341949, A341950, A341951.

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)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 3)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 2):
seq(a(n), n=2..90);  # Alois P. Heinz, Feb 24 2021

a[n_] := If[2 == n, 1, Module[{s = 0}, For[p = 2, True, p = NextPrime[p], If[p > n-p, Return[s + Boole[PrimeQ[n-1]]], s += Boole[PrimeQ[n-p]]]]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jan 03 2022, after Antti Karttunen *)

A341945(n) = if(2==n,1,my(s=0); forprime(p=2,,if(p>(n-p), return(s+isprime(n-1)), s += isprime(n-p)))); \\ Antti Karttunen, Dec 13 2021

A071335 Number of partitions of n into sum of at most three primes.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 5, 4, 5, 4, 6, 5, 6, 6, 6, 6, 7, 5, 8, 5, 8, 5, 10, 6, 8, 8, 10, 6, 11, 5, 12, 7, 12, 7, 13, 7, 14, 9, 13, 9, 15, 7, 17, 8, 15, 8, 17, 7, 17, 10, 18, 9, 20, 8, 21, 11, 21, 8, 21, 7, 23, 11, 23, 11, 23, 10, 28, 12, 25, 11, 26

Offset: 1

Reinhard Zumkeller, May 19 2002

a(n) = A010051(n) + A061358(n) + A068307(n). [From Reinhard Zumkeller, Aug 08 2009]

			a(21)=6 as 21 = 2+19 = 2+2+17 = 3+5+13 = 3+7+11 = 5+5+11 = 7+7+7.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A002375, A068307, A025583.

goldbachcount[p1_] := (parts=IntegerPartitions[p1, 3]; count=0; n=1;
  While[n<=Length[parts], If[Intersection[Flatten[PrimeQ[parts[[n]]]]][[1]]==True, count++]; n++]; count); Table[goldbachcount[i], {i, 1, 100}] (* Frank M Jackson, Mar 25 2013 *)
Table[Length[Select[IntegerPartitions[n,3],AllTrue[#,PrimeQ]&]],{n,90}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 21 2016 *)

A352231 Numbers that can be expressed as the sum of two primes in exactly 9 ways.

Original entry on oeis.org

90, 132, 170, 196, 202, 220, 230, 236, 238, 244, 250, 254, 262, 268, 302, 314, 338, 346, 356, 388, 428, 458, 488

Offset: 1

Wesley Ivan Hurt, Mar 08 2022

			90 = 7+83 = 11+79 = 17+73 = 19+71 = 23+67 = 29+61 = 31+59 = 37+53 = 43+47.

Numbers that can be expressed as the sum of two primes in k ways for k=0..10: A014092 (k=0), A067187 (k=1), A067188 (k=2), A067189 (k=3), A067190 (k=4), A067191 (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), this sequence (k=9), A352233 (k=10).

Cf. A000954, A023036, A061358.

c[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; Select[Range[1000], c[#] == 9 &] (* _Amiram Eldar, Mar 08 2022 *)

A061358(a(n)) = 9. - Alois P. Heinz, Mar 08 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

A065577 Number of Goldbach partitions of 10^n.

Original entry on oeis.org

2, 6, 28, 127, 810, 5402, 38807, 291400, 2274205, 18200488, 149091160, 1243722370, 10533150855, 90350630388

Offset: 1

Robert G. Wilson v, Dec 01 2001

Number of ways of writing 10^n as the sum of two odd primes, when the order does not matter.

			a(1)=2 because 10 = 3+7 = 5+5;
a(2)=6 because 100 = 3+97 = 11+89 = 17+83 = 29+71 = 41+59 = 47+53; ...

Ivars Peterson's MathTrek, Goldbach's Prime Pairs
Science News Online, Goldbach's Prime Pairs, week of Aug. 19, 2000; Vol. 158, No. 8.

Cf. A001031, A073610, A107318.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[n_] := Block[{c = 0, lmt = n/2, p = 3}, While[p <= lmt, If[ PrimeQ[n - p], c++ ]; p = NextPrim@p]; c]; Array[f, 10] (* Robert G. Wilson v, Nov 01 2006 *)
a[n]:=Length[Select[n - Prime[Range[PrimePi[n/2]]], PrimeQ]]; Table[a[n],{n, 10^3, 10^3}] (* Luciano Ancora, Mar 16 2015 *)

a(n) = A061358(10^n).

a(9) from Zak Seidov Nov 01 2006
a(10) from R. J. Mathar and David W. Wilson, Nov 02 2006
a(11) from David W. Wilson and Russ Cox, Nov 03 2006
a(12) from Russ Cox, Nov 04 2006
a(13) from Donovan Johnson, Nov 16 2009
a(14) from Huang Yuanbing (bailuzhou(AT)163.com), Dec 24 2009

A062301 Number of ways writing n-th prime as a sum of two primes.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Labos Elemer, Jul 05 2001

nonn

a(n) = 1 if and only if n is in A006512. - Robert Israel, Apr 04 2018

Muniru A Asiru, Table of n, a(n) for n = 1..3000

Equals A061358(A000040(n)).

Cf. A006512, A014092, A025584.

P:=Filtered([1..1000],IsPrime);; a:=List(List(List(P, i -> Partitions(i,2)), k -> Filtered(k, i -> IsPrime(i[1]) and IsPrime(i[2]))),Length); # Muniru A Asiru, Apr 05 2018

a:= n-> `if`(isprime(ithprime(n)-2), 1, 0):
seq(a(n), n=1..105);  # Alois P. Heinz, Oct 02 2020

Table[Sum[(PrimePi[Prime[n] - i] - PrimePi[Prime[n] - i - 1]) (PrimePi[i] - PrimePi[i - 1]), {i, Floor[Prime[n]/2]}], {n, 100}] (* Wesley Ivan Hurt, Apr 04 2018 *)

a(n) = isprime(prime(n) - 2) \\ David A. Corneth, Apr 04 2018

Offset changed to 1 by David A. Corneth, Apr 04 2018

A062311 Number of ways writing n! as a sum of two primes.

Original entry on oeis.org

0, 0, 0, 1, 3, 12, 39, 184, 951, 5531, 38713, 346207, 3130812, 34444964, 382437428, 4637235145

Offset: 0

Labos Elemer, Jul 05 2001

Number of unordered pairs of primes (p,q) such that n! = p + q.

			n = 4: 4! = 24 = 5+19 = 7+17 = 11+13 so a(4) = 3.

Cf. A061358, A000142. See A140088 for a variant.

In other words, a(n) = A061358(n!) = number of prime+prime partitions of n!.

a(9)-a(12) from Hans Havermann, Apr 29 2008
a(13) from Graeme McRae, May 02 2008
a(14)-a(15) from Hans Havermann, Dec 21 2008

cp's OEIS Frontend

A067187 Numbers that can be expressed as the sum of two primes in exactly one way.

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

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A100949 Number of partitions of n into a prime and a semiprime.

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A341945 Number of partitions of n into 2 primes (counting 1 as a prime).

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A071335 Number of partitions of n into sum of at most three primes.

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A352231 Numbers that can be expressed as the sum of two primes in exactly 9 ways.

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

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