A014092 -id:A014092 - cp's OEIS frontend

A066722 Numbers that can be expressed as the sum of two primes in exactly six ways.

60, 66, 72, 100, 106, 110, 116, 118, 134, 146, 166, 172, 182, 212, 248, 332

Offset: 1

Peter Bertok (peter(AT)bertok.com), Jan 13 2002

nonn

No other terms below 10000. I conjecture there are no further terms in this sequence and A067188, A067189, etc.

R. K. Guy (Jan 14 2002) remarks: "I believe that these conjectures follow from a more general one by Hardy and Littlewood (probably in Some problems of 'partitio numerorum' III, on the expression of a number as a sum of primes, Acta Math. 44(1922) 1-70)."

Index entries for sequences related to Goldbach conjecture

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), this sequence (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).

y = Select[Flatten@Table[Prime[i] + Prime[ j], {i, 500}, {j, 1, i}], # < Prime[500] &]; Select[Union[y], Count[y, #] == 6 &] (* Robert Price, Apr 22 2025 *)

A071331 Numbers having no decomposition into a sum of two prime powers.

Original entry on oeis.org

1, 149, 331, 373, 509, 701, 757, 809, 877, 907, 959, 997, 1019, 1087, 1199, 1207, 1211, 1243, 1259, 1271, 1477, 1529, 1541, 1549, 1589, 1597, 1619, 1657, 1719, 1759, 1777, 1783, 1807, 1829, 1859, 1867, 1927, 1969, 1973, 2171, 2231

Offset: 1

Reinhard Zumkeller, May 19 2002

Luca & Stanica show that this sequence contains infinitely many Fibonacci numbers. In particular, there is some N such that for all n > N, Fibonacci(1807873 + 3543120*n) is in this sequence. - Charles R Greathouse IV, Jul 06 2011

Chen shows that there are five consecutive odd numbers M-8, M-6, M-4, M-2, M, for which all are members of the sequence. Such M may be large; Chen shows that it is less than 2^(2^253000). In fact, there exists an arithmetic progression of such M, and thus they have positive density. - Charles R Greathouse IV, Jul 06 2011

T. D. Noe, Table of n, a(n) for n = 1..1000
Florian Luca and Pantelimon Stănică, Fibonacci numbers that are not sums of two prime powers, Proceedings of the American Mathematical Society 133 (2005), pp. 1887-1890.
Yong-Gao Chen, Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers, Mathematics of Computation 74 (2005), pp. 1025-1031.

A071330(a(n))=0. Cf. A000961, A109829, A014092.

a071331 n = a071331_list !! (n-1)
a071331_list = filter ((== 0) . a071330) [1..]
-- Reinhard Zumkeller, Jan 11 2013

primePowerQ[n_] := Length[FactorInteger[n]] == 1; decomposableQ[n_] := (r = False; Do[If[primePowerQ[k] && primePowerQ[n - k], r = True; Break[]], {k, 1, Floor[n/2]}]; r); Select[Range[3000], !decomposableQ[#]& ] (* Jean-François Alcover, Jun 13 2012 *)
Join[{1},Select[Range[4,2300],Count[IntegerPartitions[#,{2}],?( AllTrue[ #,PrimePowerQ]&)]==0&]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, May 28 2021 *)

isprimepower(n)=ispower(n,,&n);isprime(n)||n==1;
isA071331(n)=forprime(p=2,n\2,if(isprimepower(n-p),return(0)));forprime(p=2,sqrtint(n\2),for(e=1,log(n\2)\log(p),if(isprimepower(n-p^e),return(0))));!isprimepower(n-1)
\\ Charles R Greathouse IV, Jul 06 2011

A352229 Numbers that can be expressed as the sum of two primes in exactly 7 ways.

Original entry on oeis.org

78, 96, 112, 130, 140, 176, 178, 194, 206, 208, 218, 224, 226, 232, 272, 278, 326, 398

Offset: 1

Wesley Ivan Hurt, Mar 08 2022

			78 = 5+73 = 7+71 = 11+67 = 17+61 = 19+59 = 31+47 = 37+41.

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), this sequence (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).

c[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; Select[Range[1000], c[#] == 7 &] (* _Amiram Eldar, Mar 08 2022 *)

A352230 Numbers that can be expressed as the sum of two primes in exactly 8 ways.

Original entry on oeis.org

84, 102, 108, 138, 142, 154, 160, 184, 190, 200, 214, 242, 256, 266, 284, 292, 296, 308, 362, 368

Offset: 1

Wesley Ivan Hurt, Mar 08 2022

			84 = 5+79 = 11+73 = 13+71 = 17+67 = 23+61 = 31+53 = 37+47 = 41+43.

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), this sequence (k=8), A352231 (k=9), A352233 (k=10).

c[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; Select[Range[1000], c[#] == 8 &] (* _Amiram Eldar, Mar 08 2022 *)

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

A352233 Numbers that can be expressed as the sum of two primes in exactly 10 ways.

Original entry on oeis.org

114, 126, 162, 260, 290, 304, 316, 328, 344, 352, 358, 374, 382, 416, 542, 632

Offset: 1

Wesley Ivan Hurt, Mar 08 2022

All terms are even. Conjecture: 632 is the last term. Hardy and Littlewood conjectured a growth rate of the number of decompositions for large even numbers (see Conjecture A in page 32 of Hardy and Littlewood reference), implying this sequence is finite. - Chai Wah Wu, Mar 10 2022

			114 = 5+109 = 7+107 = 11+103 = 13+101 = 17+97 = 31+83 = 41+73 = 43+71 = 47+67 = 53+61.

G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Mathematica, volume 44, pages 1-70 (1923).

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), A352231 (k=9), this sequence (k=10).

c[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; Select[Range[1000], c[#] == 10 &] (* _Amiram Eldar, Mar 08 2022 *)

A166081 Natural numbers that are not the sum of two distinct primes.

Original entry on oeis.org

1, 2, 3, 4, 6, 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79, 83, 87, 89, 93, 95, 97, 101, 107, 113, 117, 119, 121, 123, 125, 127, 131, 135, 137, 143, 145, 147, 149, 155, 157, 161, 163, 167, 171, 173, 177, 179, 185, 187, 189, 191, 197, 203

Offset: 1

Juri-Stepan Gerasimov, Oct 06 2009

nonn

All numbers that appear in A014092 are also in this sequence, by definition.
It seems that, for n > 6, the reverse is also true, however this is unproved. - Ely Golden, Dec 25 2016
All numbers that appear in this sequence but not A014092 must be even semiprimes with no other partitions into primes. - Ely Golden, Dec 25 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from G. C. Greubel)

Cf. A117929, A006881, A038609 (complement), A014092, A066615.

Select[Range@ 204, Length@Select[Transpose@{#, Reverse@ # - 1} &@ Range[#] &@ #, Times @@ Boole@ Map[PrimeQ, #] == 1 && First@ # != Last@ # &] == 0 &] (* Michael De Vlieger, Apr 24 2016 *)
max = 1000;
ip = PrimePi[max];
A038609 = Table[Prime[i] + Prime[j], {i, ip}, {j, i + 1, ip}] // Flatten // Union // Select[#, # <= max&]&;
Complement[Range[max], A038609] (* Jean-François Alcover, Mar 24 2020 *)

{1} U A025584 U A109934. - R. J. Mathar, Oct 08 2009

A000027 \ A038609. - R. J. Mathar, Oct 14 2009

A048974 Odd numbers that are the sum of 2 primes.

Original entry on oeis.org

5, 7, 9, 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

Offset: 1

N. J. A. Sloane

A048974, A052147, A067187 and A088685 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003

Robert Price, Table of n, a(n) for n = 1..10000

Cf. A014092, A065091.

Select[Flatten@Table[Prime[i] + Prime[j], {i, 100}, {j, 1, i}], # < Prime[100] && OddQ[#] &] (* Robert Price, Apr 21 2025 *)

One of the primes must be 2, so this is simply the odd primes + 2.

a(n) = A065091(n) + 2. - Sean A. Irvine, Jul 15 2021

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

A224708 The number of unordered partitions {a,b} of n such that a and b are composite.

Original entry on oeis.org

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

Offset: 1

J. Stauduhar, Apr 16 2013

For n > 11, a(n) > 0. - Geoffrey Critzer, Jan 31 2015

Last occurrence of n is a(A014092(n+4)). - Anthony Browne, May 25 2016

			For n=8, in the set {{7,1},{6,2},{5,3},{4,4}}, {4,4} is the only partition {a,b} where a and b are both composite, so a(8)=1.
For n=12, we have partitions {8,4} and {6,6}, so a(12)=2.

J. Stauduhar, Table of n, a(n) for n = 1..10000

Cf. A010051, A014092, A045917, A066247.

nn = 76; Rest[Transpose[CoefficientList[Series[Product[1/(1 - y x^i), {i, Select[Range[2, nn], ! PrimeQ[#] &]}], {x,0,nn}], {x, y}]][[3]]] (* Geoffrey Critzer, Jan 31 2015 *)
f[n_] := Count[ PrimeQ@ Rest@ IntegerPartitions[ n, {2}], {False, False}]; Array[f, 76] (* Robert G. Wilson v, Feb 04 2015 *)

a(2*n) - a(2*n+1) + A010051(n) = A045917(n). - Anthony Browne, May 03 2016

a(A014092(n+4)) = n. - Anthony Browne, May 25 2016

cp's OEIS Frontend

A066722 Numbers that can be expressed as the sum of two primes in exactly six ways.

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A071331 Numbers having no decomposition into a sum of two prime powers.

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A352229 Numbers that can be expressed as the sum of two primes in exactly 7 ways.

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A352230 Numbers that can be expressed as the sum of two primes in exactly 8 ways.

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

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A352233 Numbers that can be expressed as the sum of two primes in exactly 10 ways.

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A166081 Natural numbers that are not the sum of two distinct primes.

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A048974 Odd numbers that are the sum of 2 primes.

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

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