A002375 -id:A002375 - cp's OEIS frontend

A000974 Conjecturally the number of even integers the sum of two primes in exactly n ways.

1, 4, 9, 11, 11, 16, 16, 18, 20, 23, 16, 29, 16, 25, 27, 23, 22, 25, 35, 29, 26, 25, 27, 27, 27, 33, 28, 44, 35, 21, 29, 35, 38, 33, 39, 37, 34, 35, 31, 31, 28, 41, 37, 32, 44, 35, 37, 41, 44, 33, 37, 32, 47, 39, 43, 47, 33, 37, 48, 41, 37, 48, 34, 35, 47, 36, 29, 36, 46, 44, 43, 38, 48

Offset: 0

Bill Gosper

nonn

T. D. Noe, Table of n, a(n) for n = 0..5000

Cf. A000954, A001172, A002375.

f[n_] := Length[ Select[2n - Prime[ Range[ PrimePi[n]]], PrimeQ]]; a = Table[0, {75}]; Do[c = f[n]; If[c < 75, a[[c + 1]]++ ], {n, 5000}]; a (* Robert G. Wilson v, using Paul Abbott's coding in A002375, Apr 07 2005 *)

A002092 From a Goldbach conjecture: records in A185091.

Original entry on oeis.org

1, 3, 5, 7, 17, 29, 47, 61, 73, 83, 277, 317, 349, 419, 503, 601, 709, 829, 877, 1129, 1237, 1367, 1429, 1669, 1801, 2467, 2833, 2879, 3001, 3037, 3329, 3821, 4861, 5003, 5281, 5821, 5897, 6301, 6329, 6421, 6481, 6841, 7069, 7121, 7309, 7873, 8017, 8597, 8821

Offset: 1

N. J. A. Sloane

nonn

See A002091. The sequence gives the record values of q in the representations minimizing q of 2*k+1 = 2*p+q, p prime, q {1,prime}.

Checked up to 2*k = 10^13.

Brian H. Mayoh, On the second Goldbach conjecture. Nordisk Tidskr. Informations-Behandling 6, 1966, pp. 48-50.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

See A002091.
Index entries for sequences related to Goldbach conjecture

Cf. A002372, A002373, A002374, A002375, A045917, A006307, A002091, A185091, A194828, A194829.

a(n) = A185091((A002091(n)+1)/2).

Comment added, a(19)-a(32) from Hugo Pfoertner, Sep 03 2011
a(33) from Jason Kimberley, a(34)-a(40) from Hugo Pfoertner, Sep 09 2011
a(41)-a(49) from Hugo Pfoertner, Sep 25 2011

A024684 Number of ways prime(n) is a sum of three distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 6, 9, 11, 13, 13, 15, 18, 17, 19, 25, 26, 28, 32, 31, 37, 32, 39, 44, 51, 54, 48, 60, 55, 62, 64, 73, 80, 79, 74, 89, 84, 92, 86, 92, 107, 119, 105, 118, 122, 117, 135, 143, 146, 147, 141, 149, 159, 157, 176, 175, 190, 180

Offset: 1

Clark Kimberling

nonn

			a(9) = 2 because prime(9) = 23 and 23 = 3 + 7 + 13 = 5 + 7 + 11.

Zak Seidov and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 2000 terms from Zak Seidov)
Wikipedia, Goldbach's weak conjecture

Cf. A002372, A002375, A224534.

last = 313; pp = PrimePi[last]; t = Select[Sort[Tally[Select[Total /@ Subsets[Prime[Range[2, pp]], {3}], PrimeQ]]], #[[1]] <= last &]; Join[{0, 0, 0, 0, 0, 0, 0}, Transpose[t][[2]]] (* T. D. Noe, Apr 15 2013 *)

a(n) = A125688(prime(n)). - R. J. Mathar, Jun 09 2014

A053033 Numbers which are the average of two primes in more ways than any smaller number.

Original entry on oeis.org

1, 2, 5, 11, 17, 24, 30, 39, 42, 45, 57, 60, 84, 90, 105, 150, 165, 195, 210, 255, 315, 390, 420, 495, 525, 570, 630, 735, 825, 840, 945, 1050, 1155, 1365, 1575, 1785, 1995, 2100, 2205, 2310, 2625, 2730, 3045, 3255, 3465, 3990, 4095

Offset: 1

Len Smiley, Feb 23 2000

nonn

From Ahmad J. Masad, Dec 09 2019: (Start)
Conjecture 1: This sequence is infinite.
Conjecture 2: If this sequence is infinite, then for each prime number p > 2, there exists a minimum sufficiently large number k such that all terms >= k are multiples of p. (End)
Apparently, all terms >= 90 are multiples of 15. - Hugo Pfoertner, Dec 09 2019
Positions of records in A045917. - Sean A. Irvine, Dec 04 2021

			a(1) = 1: average of 0 pairs of primes;
a(2) = 2: average of 1 pair of primes (2,2);
a(3) = 5: average of 2 pairs of primes (3,7), (5,5);
a(4) = 11: average of 3 pairs of primes (3,19), (5,17), (11,11);
a(5) = 17: average of 4 pairs of primes (3,31), (5,29), (11,23), (17,17).

Giovanni Resta, Table of n, a(n) for n = 1..266 (terms < 10^7)

Cf. A002375, A045917.

(for n>0): printlevel := -1:maxx := 0:for j from 2 to 1000 do count := 0; for k from 0 to j-2 do if (isprime(j-k) and isprime(j+k)) then count := count+1 fi od; if count>maxx then print(j,count); maxx := count fi od;

More terms from James Sellers, Feb 25 2000

A082917 Numbers that can be expressed as the sum of two odd primes in more ways than any smaller even number.

Original entry on oeis.org

6, 10, 22, 34, 48, 60, 78, 84, 90, 114, 120, 168, 180, 210, 300, 330, 390, 420, 510, 630, 780, 840, 990, 1050, 1140, 1260, 1470, 1650, 1680, 1890, 2100, 2310, 2730, 3150, 3570, 3990, 4200, 4410, 4620, 5250, 5460, 6090, 6510, 6930, 7980, 8190, 9030, 9240

Offset: 1

Hugo Pfoertner, Apr 15 2003

nonn

The terms up to 114 are identical with A001172. The record-setting number of decompositions is given by A082918.

It appears that every primorial number (A002110) greater than 30 is in this sequence. Sequence A116979 gives the number of decompositions for n equal to a primorial number. - T. D. Noe, Mar 15 2010

			a(1) = 6 = 3 + 3.
a(2) = 10 because 10 is the smallest number that can be written in two ways: 10 = 3 + 7 = 5 + 5.

Bill Hannaford, Table of n, a(n) for n = 1..420 (first 244 terms from T. D. Noe)

Cf. A002375, A001172, A082918. A109679 is another version of the same sequence.

kmax = 40000;
ip[k_] := IntegerPartitions[k, {2}, Select[Range[3, k-1], PrimeQ]];
seq = Module[{k, lg, record = 0, n = 0}, Reap[For[k = 6, k <= kmax, k = k+2, lg = Length[ip[k]]; If[lg > record, record = lg; n = n+1; Print["a(", n, ") = ", k]; Sow[k]]]][[2, 1]]] (* Jean-François Alcover, Jun 04 2022 *)

A224534 Prime numbers that are the sum of three distinct prime numbers.

Original entry on oeis.org

19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307

Offset: 1

T. D. Noe, Apr 15 2013

nonn

Similar to Goldbach's weak conjecture.
Primes in A124867, and by the comment in A124867 also the set of all primes >=19. - R. J. Mathar, Apr 19 2013
"Goldbach's original conjecture (sometimes called the 'ternary' Goldbach conjecture), written in a June 7, 1742 letter to Euler, states 'at least it seems that every number that is greater than 2 is the sum of three primes' (Goldbach 1742; Dickson 2005, p. 421). Note that here Goldbach considered the number 1 to be a prime, a convention that is no longer followed." [Weisstein] - Jonathan Vos Post, May 15 2013

			19 = 3 + 5 + 11.

H. A. Helfgott and David J. Platt, Numerical Verification of the Ternary Goldbach Conjecture up to 8.875e30, arXiv:1305.3062 [math.NT], 2013-2014.
H. A. Helfgott and David J. Platt, Numerical verification of the Ternary Goldbach Conjecture up to 8.875*10^30, Exp. Math. 22 (4) (2013) 406-409.
Eric W. Weisstein, Goldbach conjecture
Wikipedia, Goldbach's conjecture
Wikipedia, Goldbach's weak conjecture

Cf. A002372, A002375, A024684 (number of sums), A224535, A166063, A166061, A071621.

Union[Select[Total /@ Subsets[Prime[Range[2, 30]], {3}], PrimeQ]]

A227909 Number of ways to write 2n = p + q with p, q and (p-1)(q+1) - 1 all prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 3, 1, 2, 5, 2, 3, 2, 3, 3, 5, 3, 1, 5, 4, 5, 4, 3, 4, 7, 4, 4, 2, 1, 4, 9, 2, 4, 11, 4, 2, 6, 2, 6, 11, 6, 4, 3, 3, 5, 6, 4, 3, 6, 2, 4, 10, 3, 10, 12, 7, 1, 6, 6, 5, 11, 4, 5, 6, 4, 3, 11, 2, 10, 13, 4, 6, 5, 2, 14, 13, 2, 2, 5, 5, 9, 15, 5, 3, 7, 8, 5, 3, 5, 7, 15, 3, 1, 8, 5, 7, 11, 4

Offset: 1

Olivier Gérard and Zhi-Wei Sun, Oct 13 2013

nonn

Conjecture: a(n) > 0 for all n > 1.
This is stronger than Goldbach's conjecture for even numbers. It also implies A. Murthy's conjecture (cf. A109909) for even numbers.
We have verified the conjecture for n up to 2*10^7.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023

			a(6) = 1 since 2*6 = 5 + 7, and (5-1)*(7+1)-1 = 31 is prime.
a(10) = 1 since 2*10 = 7 + 13, and (7-1)*(13+1)-1 = 83 is prime.
a(20) = 1 since 2*20 = 17 + 23, and (17-1)*(23+1)-1 = 383 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A002375, A109909, A218754, A219052, A220431, A220455, A220554, A227908, A230230.

a[n_]:=Sum[If[PrimeQ[2n-Prime[i]]&&PrimeQ[(Prime[i]-1)(2n-Prime[i]+1)-1],1,0],{i,1,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]

A230224 Number of ways to write 2*n = p + q + r + s with p <= q <= r <= s such that p, q, r, s are primes in A230223.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 3, 3, 3, 4, 4, 3, 5, 4, 5, 3, 6, 4, 6, 5, 5, 5, 7, 5, 9, 4, 6, 6, 8, 6, 9, 5, 7, 7, 10, 6, 8, 7, 8, 7, 9, 5, 10, 7, 11, 7, 7, 7, 11, 7, 10, 6, 10, 6, 13, 7, 9, 7, 11, 9, 11, 7, 9, 6, 14, 8, 12, 6, 13, 11, 12, 11, 13, 10, 16, 9, 14, 7, 14

Offset: 1

Zhi-Wei Sun, Oct 12 2013

nonn

Conjecture: a(n) > 0 for all n > 17.

			a(21) = 1 since 2*21 = 7 + 7 + 11 + 17, and 7, 11, 17 are primes in A230223.
a(27) = 1 since 2*27 = 7 + 11 + 17 + 19, and 7, 11, 17, 19 are primes in A230223.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A002375, A068307, A230223, A230140, A230141, A230219.

RQ[n_]:=n>5&&PrimeQ[3n-4]&&PrimeQ[3n-10]&&PrimeQ[3n-14]
SQ[n_]:=PrimeQ[n]&&RQ[n]
a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&RQ[Prime[k]]&&SQ[2n-Prime[i]-Prime[j]-Prime[k]],1,0],
{i,1,PrimePi[n/2]},{j,i,PrimePi[(2n-Prime[i])/3]},{k,j,PrimePi[(2n-Prime[i]-Prime[j])/2]}]
Table[a[n],{n,1,100}]

A230230 Number of ways to write 2n = p + q with p, q, 3p - 10, 3*q + 10 all prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 3, 5, 1, 5, 5, 3, 4, 5, 3, 2, 6, 4, 3, 6, 3, 3, 6, 3, 5, 6, 3, 6, 5, 4, 4, 9, 5, 4, 9, 5, 3, 9, 4, 4, 6, 4, 5, 6, 5, 5, 10, 4, 8, 10, 3, 7, 12, 3, 6, 11, 5, 7, 8, 3, 4, 6, 6, 4, 7, 2, 7, 9, 2, 10, 9, 3, 9, 8, 3, 5, 14, 8, 4, 12, 5, 5, 11, 5, 6, 8, 3, 8, 7, 4, 9, 11, 3

Offset: 1

Zhi-Wei Sun, Oct 12 2013

nonn

Conjecture: a(n) > 0 for all n > 3.
This is stronger than Goldbach's conjecture for even numbers. If 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime, then 6*n is the sum of the two primes 3*p - 10 and 3*q + 10.
Conjecture verified for 2*n up to 10^9. - Mauro Fiorentini, Jul 08 2023

			a(5) = 1 since 2*5 = 7 + 3 with 3*7 - 10 = 11 and 3*3 + 10 = 19 both prime.
a(16) = 1 since 2*16 = 13 + 19 with 3*13 - 10 = 29 and 3*19 + 10 = 67 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A002375, A187757, A199920, A227923, A230227, A230140, A230141, A230217, A230219, A230223, A230224.

PQ[n_]:=n>3&&PrimeQ[3n-10]
SQ[n_]:=PrimeQ[n]&&PrimeQ[3n+10]
a[n_]:=Sum[If[PQ[Prime[i]]&&SQ[2n-Prime[i]],1,0],{i,1,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]

A233654 |{prime p < n: n - p = sigma(k) for some k > 0}|, where sigma(k) is the sum of all (positive) divisors of k.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 14 2013

nonn

Conjecture: (i) Let n > 1 be an integer. Then we have a(2*n) > 0. Also, 2*n + 1 can be written as p + sigma(k), where p is a Sophie Germain prime and k is a positive integer.
(ii) Each odd number greater than one can be written as sigma(k^2) + phi(m), where k and m are positive integers, and phi(.) is Euler's totient function.
That a(2*n+1) > 0 for n > 1 is a consequence of Goldbach's conjecture, for, if 2*n = p + q with p and q both prime, then 2*n + 1 = p + sigma(q) = q + sigma(p).

			a(3) = 1 since 3 = 2 + 1 = 2 + sigma(1) with 2 prime.
a(7) = 1 since 7 = 3 + 4 = 3 + sigma(3) with 3 prime.
a(10) = 3 since 10 = 2 + 8 = 2 + sigma(7) with 2 prime, 10 = 3 + 7 = 3 + sigma(4) with 3 prime, and 10 = 7 + 3 = 7 + sigma(2) with 7 prime.
a(13) = 2 since 13 = 5 + 8 = 5 + sigma(7) with 5 prime, and 13 = 7 + 6 = 7 + sigma(5) with 7 prime.
a(28) = 1 since 28 = 13 + 15 = 13 + sigma(8) with 13 prime.
a(36) = 3 since 36 = 5 + 31 = 5 + sigma(16) = 5 + sigma(25) with 5 prime, 36 = 23 + 13 = 23 + sigma(9) with 23 prime, and 36 = 29 + 7 = 29 + sigma(4) with 29 prime.
a(148) = 1 since 148 = 109 + 39 = 109 + sigma(18) with 109 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000203, A002372, A002375, A005384, A232270, A233542, A233544, A233547, A233549.

f[n_]:=Sum[If[Mod[n,d]==0,d,0],{d,1,n}]
S[n_]:=Union[Table[f[j],{j,1,n}]]
PQ[n_]:=n>0&&PrimeQ[n]
a[n_]:=Sum[If[PQ[n-Part[S[n],i]],1,0],{i,1,Length[S[n]]}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A000974 Conjecturally the number of even integers the sum of two primes in exactly n ways.

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A002092 From a Goldbach conjecture: records in A185091.

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A024684 Number of ways prime(n) is a sum of three distinct primes.

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A053033 Numbers which are the average of two primes in more ways than any smaller number.

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A082917 Numbers that can be expressed as the sum of two odd primes in more ways than any smaller even number.

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A224534 Prime numbers that are the sum of three distinct prime numbers.

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A227909 Number of ways to write 2*n = p + q with p, q and (p-1)*(q+1) - 1 all prime.

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A230224 Number of ways to write 2*n = p + q + r + s with p <= q <= r <= s such that p, q, r, s are primes in A230223.

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A227909 Number of ways to write 2n = p + q with p, q and (p-1)(q+1) - 1 all prime.

A230230 Number of ways to write 2n = p + q with p, q, 3p - 10, 3*q + 10 all prime.