A352231 -id:A352231 - cp's OEIS frontend

A014092 Numbers that are not the sum of 2 primes.

1, 2, 3, 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, 205, 207, 209

Offset: 1

N. J. A. Sloane

Suggested by the Goldbach conjecture that every even number larger than 2 is the sum of 2 primes.
Since (if we believe the Goldbach conjecture) all the entries > 2 in this sequence are odd, they are equal to 2 + an odd composite number (or 1).
Otherwise said, the sequence consists of 2 and odd numbers k such that k-2 is not prime. In particular there is no element from A006512, greater of a twin prime pair. - M. F. Hasler, Sep 18 2012
Values of k such that A061358(k) = 0. - Emeric Deutsch, Apr 03 2006
Values of k such that A073610(k) = 0. - Graeme McRae, Jul 18 2006

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 2.8 (for Goldbach conjecture).

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

Cf. A002372, A002373, A002374, A048974, A061358.
Cf. A010051, A000040, A051035 (composites).
Equivalent sequence for prime powers: A071331.
Numbers that can be expressed as the sum of two primes in k ways for k=0..10: this sequence (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), A352233 (k=10).

a014092 n = a014092_list !! (n-1)
a014092_list = filter (\x ->
   all ((== 0) . a010051) $ map (x -) $ takeWhile (< x) a000040_list) [1..]
-- Reinhard Zumkeller, Sep 28 2011

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j),j=1..50): gser:=series(g,x=0,230): a:=proc(n) if coeff(gser,x^n)=0 then n else fi end: seq(a(n),n=1..225); # Emeric Deutsch, Apr 03 2006

s1falsifiziertQ[s_]:= Module[{ip=IntegerPartitions[s, {2}], widerlegt=False},Do[If[PrimeQ[ip[[i,1]] ] ~And~ PrimeQ[ip[[i,2]] ], widerlegt = True; Break[]],{i,1,Length[ip]}];widerlegt]; Select[Range[250],s1falsifiziertQ[ # ]==False&] (* Michael Taktikos, Dec 30 2007 *)
Join[{1,2},Select[Range[3,300,2],!PrimeQ[#-2]&]] (* Zak Seidov, Nov 27 2010 *)
Select[Range[250],Count[IntegerPartitions[#,{2}],?(AllTrue[#,PrimeQ]&)]==0&] (* _Harvey P. Dale, Jun 08 2022 *)

isA014092(n)=local(p,i) ; i=1 ; p=prime(i); while(pA014092(a), print(n," ",a); n++)) \\ R. J. Mathar, Aug 20 2006

from sympy import prime, isprime
def ok(n):
    i=1
    x=prime(i)
    while xIndranil Ghosh, Apr 29 2017

Odd composite numbers + 2 (essentially A014076(n) + 2 ).

Equals {2} union A005408 \ A052147, i.e., essentially the complement of A052147 (or rather A048974) within the odd numbers A005408. - M. F. Hasler, Sep 18 2012

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

Original entry on oeis.org

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

A067188 Numbers that can be expressed as the (unordered) sum of two primes in exactly two ways.

Original entry on oeis.org

10, 14, 16, 18, 20, 28, 32, 38, 68

Offset: 1

Amarnath Murthy, Jan 10 2002

Corresponds to numbers 2m such that A045917(m)=2. Subsequence of A014091. - Lekraj Beedassy, Apr 22 2004

			18 is a term as 18 = 13+5 = 11+7 are the only two ways to express 18 as a sum of two primes.

Index entries for sequences related to Goldbach conjecture

Cf. A023036.

Numbers that can be expressed as the sum of two primes in k ways for k=0..10: A014092 (k=0), A067187 (k=1), this sequence (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).

y = Select[Flatten@Table[Prime[i] + Prime[j], {i, 100}, {j, 1, i}], # < Prime[100] &]; Select[Union[y], Count[y, #] == 2 &] (* Robert Price, Apr 22 2025 *)

Corrected by Peter Bertok (peter(AT)bertok.com), who finds (Jan 13 2002) that there are no other terms below 10000 and conjectures there are no further terms in this sequence and A067189, A067190, etc.

R. K. Guy (Jan 14 2002) remarks: "I believe that these conjectures follow from a more general one by Hardy & 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)."

A067189 Numbers that can be expressed as the sum of two primes in exactly three ways.

Original entry on oeis.org

22, 24, 26, 30, 40, 44, 52, 56, 62, 98, 128

Offset: 1

Amarnath Murthy, Jan 10 2002

Corresponds to numbers 2m such that A045917(m)=3. Subsequence of A014091. - Lekraj Beedassy, Apr 22 2004

			26 is a term as 26 = 23+3 = 19+7 = 13+13 are all the three ways to express 26 as a sum of two primes.

Index entries for sequences related to Goldbach conjecture

Cf. A023036.

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), this sequence (k=3), A067190 (k=4), A067191 (k=5), A066722 (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, #] == 3 &] (* Robert Price, Apr 22 2025 *)

Extended by Peter Bertok (peter(AT)bertok.com), who finds (Jan 13 2002) that there are no other terms below 10000 and conjectures there are no further terms in this sequence and A067188, A067190, etc.

R. K. Guy (Jan 14 2002) remarks: "I believe that these conjectures follow from a more general one by Hardy & 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)."

A067190 Numbers that can be expressed as the sum of two primes in exactly four ways.

Original entry on oeis.org

34, 36, 42, 46, 50, 58, 80, 88, 92, 122, 152

Offset: 1

Amarnath Murthy, Jan 10 2002

			36 is a term as 36 = 31 + 5 = 29 + 7 = 23 + 13 = 19 + 17 are all the four ways to express 36 as a sum of two primes.

Index entries for sequences related to Goldbach conjecture

Cf. A023036.

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), this sequence (k=4), A067191 (k=5), A066722 (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, #] == 4 &] (* Robert Price, Apr 22 2025 *)

Extended by Peter Bertok (peter(AT)bertok.com), who finds (Jan 13 2002) that there are no other terms below 10000 and conjectures 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)."

A067191 Numbers that can be expressed as the sum of two primes in exactly five ways.

Original entry on oeis.org

48, 54, 64, 70, 74, 76, 82, 86, 94, 104, 124, 136, 148, 158, 164, 188

Offset: 1

Amarnath Murthy, Jan 10 2002

There are no other terms below 10000 and I conjecture there are no further terms in this sequence and A067188, A067189, etc. - Peter Bertok (peter(AT)bertok.com), Jan 13 2002
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). - R. K. Guy, Jan 14 2002
There are no further terms through 50000. - David Wasserman, Jan 15 2002

			70 is a term as 70 = 67 + 3 = 59 + 11 = 53 + 17 = 47 + 23 41 + 29 are all the five ways to express 70 as a sum of two primes.

Index entries for sequences related to Goldbach conjecture

Cf. A002375, A023036.

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), this sequence (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).

upperbound=10^4; range=ConstantArray[0,2*upperbound];
primeRange=Prime[Range[PrimePi[upperbound]]];
(range[[Plus@@#]]++)&/@(DeleteDuplicates[Sort[#]&/@Tuples[primeRange,2]]);{"upperbound="<>ToString[upperbound],Flatten[Position[Take[range,upperbound],5]]} (* Hans Rudolf Widmer, Jul 06 2021 *)

Corrected and extended by Peter Bertok (peter(AT)bertok.com), Jan 13 2002

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

Original entry on oeis.org

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 *)

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 *)

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 *)

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A014092 Numbers that are not the sum of 2 primes.

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

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A067188 Numbers that can be expressed as the (unordered) sum of two primes in exactly two ways.

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A067189 Numbers that can be expressed as the sum of two primes in exactly three ways.

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A067190 Numbers that can be expressed as the sum of two primes in exactly four ways.

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A067191 Numbers that can be expressed as the sum of two primes in exactly five ways.

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

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