A045917 -id:A045917 - cp's OEIS frontend

A226237 Sum of the parts in the Goldbach partitions of 2n.

0, 4, 6, 8, 20, 12, 28, 32, 36, 40, 66, 72, 78, 56, 90, 64, 136, 144, 76, 120, 168, 132, 184, 240, 200, 156, 270, 168, 232, 360, 186, 320, 396, 136, 350, 432, 370, 380, 546, 320, 410, 672, 430, 352, 810, 368, 470, 672, 294, 600, 816, 520, 636, 864, 660, 784

Offset: 1

Wesley Ivan Hurt, Aug 25 2013

Goldbach's Conjecture states that every positive even integer > 4 is expressible as the sum of two odd primes in at least one way. This is logically equivalent to the statement that a(n) > 0 for n > 2.

The sum of the parts in the partitions of 2n into exactly two prime parts.

			a(13) = 78.  Since 2*13 = 26 has exactly 3 Goldbach partitions: (23,3),(19,7), and (13,13).  The sum of the parts gives: 23+19+13+13+7+3 = 78.

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A045917, A185297, A187129, A187619 (Sum of differences).

with(numtheory); A226237:=n->2*n*sum( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)), i=1..n); seq(A226237(n), n=1..100);

Table[ 2 n*Sum[ Floor[2/PrimeOmega[2 n*i - i^2]], {i, 2, n}], {n,
  100}]

a(n) = 2n * A045917(n). a(n) = A185297(n) + A187129(n), n>1.

A228553 Sum of the products formed by multiplying together the smaller and larger parts of each Goldbach partition of 2n.

Original entry on oeis.org

0, 4, 9, 15, 46, 35, 82, 94, 142, 142, 263, 357, 371, 302, 591, 334, 780, 980, 578, 821, 1340, 785, 1356, 1987, 1512, 1353, 2677, 1421, 2320, 4242, 1955, 2803, 4362, 1574, 4021, 5298, 4177, 4159, 6731, 4132, 5593, 9808

Offset: 1

Wesley Ivan Hurt, Aug 25 2013

nonn

Since the product of each prime pair is semiprime and since we are adding A045917(n) of these, a(n) is expressible as the sum of exactly A045917(n) distinct semiprimes.

			a(5) = 46. 2*5 = 10 has two Goldbach partitions: (7,3) and (5,5). Taking the products of the larger and smaller parts of these partitions and adding, we get 7*3 + 5*5 = 46.

Cf. A010051, A045917, A064911, A105020, A185297, A187129.

with(numtheory); seq(sum( (2*k*i-i^2) * (pi(i)-pi(i-1)) * (pi(2*k-i)-pi(2*k-i-1)),  i=2..k), k=1..70);
# Alternative:
f:= proc(n)
  local S;
  S:= select(t -> isprime(t) and isprime(2*n-t), [seq(i,i=3..n,2)]);
  add(t*(2*n-t),t=S)
end proc:
f(2):= 4:
map(f, [$1..200]); # Robert Israel, Nov 29 2020

c[n_] := Boole[PrimeQ[n]];
a[n_] := Sum[c[i]*c[2n-i]*i*(2n-i), {i, 2, n}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 02 2023 *)

a(n) = Sum_{i=2..n} c(i) * c(2*n-i) * i * (2*n-i), where c = A010051.

a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} c(A105020(k)) * A105020(k), where c = A064911. - Wesley Ivan Hurt, Sep 19 2021

A238134 Number of primes p < n with q = floor((n-p)/4) and prime(q) - q + 1 both prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 03 2014

nonn

Conjecture: Let m > 0 and n > 2*m + 1 be integers. If m = 1 and 2 | n, or m = 3 and n is not congruent to 1 modulo 6, or m = 2, 4, 5, ..., then there is a prime p < n such that q = floor((n-p)/m) and prime(q) - q + 1 are both prime.
In the cases m = 1, 2, this gives refinements of Goldbach's conjecture and Lemoine's conjecture (see also A235189). For m > 2, the conjecture is completely new.
See also A238701 for a similar conjecture involving primes q with q^2 - 2 also prime.

			 a(29) = 3 since 7, floor((29-7)/4) = 5 and prime(5) - 5 + 1 = 11 - 4 = 7 are all prime; 17, floor((29-17)/4) = 3 and prime(3) - 3 + 1 = 5 - 2 = 3 are all prime; 19, floor((29-19)/4) = 2 and prime(2) - 2 + 1 = 3 - 1 = 2 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A045917, A046927, A234694, A234695, A235189, A237705, A238701.

PQ[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]
a[n_]:=Sum[If[PQ[Floor[(n-Prime[k])/4]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

A238701 Number of primes p < n with q = floor((n-p)/4) and q^2 - 2 both prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 03 2014

nonn

Conjecture: Let m > 0 and n > 2*m + 1 be integers. If m = 1 and 2 | n, or m = 3 and n is not congruent to 1 modulo 6, or m = 2, 4, 5, ..., then there is a prime p < n with q = floor((n-p)/m) and q^2 - 2 both prime.

In the case m = 1, this is a refinement of Goldbach's conjecture. In the case m = 2, this is stronger than Lemoine's conjecture (cf. A046927). The conjecture for m > 2 seems completely new. We view the conjecture as a natural extension of Goldbach's conjecture.

			a(11) = 2 since 2, floor((11-2)/4)= 2 and 2^2 - 2 are all prime, and 3, floor((11-3)/4) = 2 and 2^2 - 2 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A045917, A046927, A049002, A062326, A230502, A237413, A237705.

PQ[n_]:=PrimeQ[n]&&PrimeQ[n^2-2]
p[n_,k_]:=PQ[Floor[(n-Prime[k])/4]]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

A258713 A001172(n)/2: Least k such that 2k is a sum of two odd primes in exactly n ways.

Original entry on oeis.org

0, 3, 5, 11, 17, 24, 30, 39, 42, 45, 57, 72, 60, 84, 90, 117, 123, 144, 120, 105, 162, 150, 180, 237, 165, 264, 288, 195, 231, 240, 210, 285, 255, 336, 396, 378, 438, 357, 399, 345, 519, 315, 504, 465, 390, 480, 435, 462, 450, 567, 717, 420, 495, 651

Offset: 0

N. J. A. Sloane, Jun 14 2015

nonn

Up to a(14) also indices of records in A002375, number of ways to write 2n as sum of two odd primes. - M. F. Hasler, Aug 21 2017

Robert Israel, Table of n, a(n) for n = 0..845

Cf. A136244, A001172.

Cf. A002375, A002372, A045917.

g:= add(x^ithprime(i),i=2..1000):
G:= series((g^2+add(x^(2*ithprime(i)),i=2..1000))/2,x,ithprime(1001)+3):
A[0]:= 0:
for k from 1 to (ithprime(1001)+1)/2 do
  m:= coeff(G,x,2*k);
  if not assigned(A[m]) then A[m]:= k fi;
od:
for m from 1 while assigned(A[m]) do od:
seq(A[i],i=0..m-1); # Robert Israel, Aug 21 2017

With[{s = Array[Count[Select[IntegerPartitions[2 #, 2], Length@ # == 2 &], p_ /; AllTrue[p, And[PrimeQ@ #, OddQ@ #] &]] &, 10^3]}, Table[FirstPosition[s, n][[1]] /. 1 -> 0, {n, 0, 53}]] (* Michael De Vlieger, Aug 21 2017 *)

Edited by M. F. Hasler, Aug 21 2017

Edited by Robert Israel, Aug 21 2017

A279481 Count the primes appearing in each interval [p,q] where (p,q) is a Goldbach partition of 2n, and then add the results.

Original entry on oeis.org

0, 0, 1, 2, 4, 2, 5, 8, 6, 9, 13, 12, 14, 10, 12, 12, 24, 22, 9, 20, 24, 27, 29, 38, 36, 24, 39, 29, 33, 43, 24, 58, 58, 17, 52, 60, 53, 63, 80, 46, 54, 87, 70, 46, 100, 62, 58, 87, 31, 79, 104, 71, 87, 119, 99, 116, 152, 114, 94, 181, 54, 82, 144, 39, 116, 133

Offset: 1

Wesley Ivan Hurt, Dec 12 2016

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A000720, A010051, A045917, A278700, A279103.

with(numtheory): A279481:=n->add( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (pi(2*n-i)-pi(i-1)), i=2..n): 0,0,seq(A279481(n), n=3..100);

f[n_] := Sum[ Boole[ PrimeQ[ i]] Boole[ PrimeQ[ 2n -i]] (PrimePi[ 2n -i] - PrimePi[i -1]), {i, 2, n}]; f[2] = 0; Array[ f, 80] (* Robert G. Wilson v, Dec 15 2016 *)

a(n) = Sum_{i=2..n} A010051(i)*A010051(2*n-i)*(pi(2*n-i)-pi(i-1)) for n > 2.

A281687 Number of partitions of 2*n into the sum of two totient numbers (A002202).

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Jan 27 2017

See also graph of A045917 ("Goldbach's comet"). - Altug Alkan, Jan 30 2017

			a(6) = 3 because 2 * 6 = 12 = 2 + 10 = 4 + 8 = 6 + 6 and 2, 4, 6, 8, 10 are in A002202.

Robert Israel, Table of n, a(n) for n = 1..10000
Altug Alkan, Illustration Of Residue Classes Modulo 6

Cf. A000010, A002202, A045917, A280867.

N:= 1000: V:= Vector(2*N):
V[1]:= 1:
for n from 2 to 2*N by 2 do
  if nops(numtheory:-invphi(n))>1 then V[n]:= 1 fi
od:
C:= map(round,SignalProcessing:-Convolution(V,V)):
seq((C[2*i-1]+V[i])/2,i=1..N); # Robert Israel, Jan 27 2017

a(n) = sum(k=1, n, istotient(k) && istotient(2*n-k));

A325142 a(n) = k if (n - k, n + k) is the centered Goldbach partition of 2n if it exists and -1 otherwise.

Original entry on oeis.org

-1, -1, 0, 0, 1, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 9, 0, 5, 6, 3, 4, 9, 0, 1, 0, 9, 4, 3, 6, 5, 0, 9, 2, 3, 0, 1, 0, 3, 2, 15, 0, 5, 12, 3, 8, 9, 0, 7, 12, 3, 4, 15, 0, 1, 0, 9, 4, 3, 6, 5, 0, 15, 2, 3, 0, 1, 0, 15, 4, 3, 6, 5, 0, 9, 2, 15, 0

Offset: 0

Peter Luschny, May 02 2019

sign

Let N = 2*n = p + q where p and q are primes. We call such a pair (p, q) a Goldbach partition of N. A centered Goldbach partition is the Goldbach partition of the form (n - k, n + k) where k >= 0 is minimal. If N has a centered Goldbach partition then a(n) is this k and otherwise -1.

According to Goldbach's conjecture, any even N = 2n > 2 has a Goldbach partition, which is necessarily of the form p = n - k, q = n + k: namely, with n = (p+q)/2 and k = (q-p)/2. - M. F. Hasler, May 02 2019

			a(162571) = 78 because 325142 = 162493 + 162649 and there is no k, 0 <= k < 78, such that (162571 - k, 162571 + k) is a Goldbach partition of 325142.

Peter Luschny, Table of n, a(n) for n = 0..10000

Cf. A045917, A047160, A066285.

Cf. A002374, A112823, A002374, A234345, A020482.

a := proc(n) local k; for k from 0 to n do
if isprime(n + k) and isprime(n - k) then return k fi od: -1 end:
seq(a(n), n=0..83);

a[n_] := Module[{k}, For[k = 0, k <= n, k++, If[PrimeQ[n+k] && PrimeQ[n-k], Return[k]]]; -1]; Table[a[n], {n, 0, 83}] (* Jean-François Alcover, Jul 06 2019, from Maple *)

a(n) = for(k=0, n, if(ispseudoprime(n+k) && ispseudoprime(n-k), return(k))); -1 \\ Felix Fröhlich, May 02 2019

apply( A325142(n)=-!forprime(p=n,2*n, isprime(n*2-p)&&return(p-n)), [0..99]) \\ M. F. Hasler, May 02 2019

a(n) = n - A112823(n) = A234345(n) - n (= n - A002374(n) for n > 2). - M. F. Hasler, May 02 2019

a(n) = A047160(n) = A066285(n)/2 for n >= 2. - Alois P. Heinz, Jun 01 2020

A362640 Product of the larger primes, q, in the Goldbach partitions of 2n such that p + q = 2n, p <= q, and p,q prime (or 1 if no Goldbach partition of 2n exists).

Original entry on oeis.org

1, 2, 3, 5, 35, 7, 77, 143, 143, 221, 3553, 4199, 5681, 391, 7429, 551, 351509, 392863, 589, 24679, 765049, 47027, 1175921, 58642669, 2318087, 55883, 95041567, 84323, 2961799, 5037203051, 78647, 367569469, 14263488419, 2257, 403723843, 22531226387, 461671607, 761740327

Offset: 1

Wesley Ivan Hurt, Apr 28 2023

			a(10) = 221; 2*10 = 20 has two Goldbach partitions, namely 17+3 and 13+7. The product of the larger parts of these partitions, is 17*13 = 221.

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A010051, A045917, A337568 (product of all prime parts), A362641 (product of smaller primes p).

Table[Product[(2 n - k)^((PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1])), {k, n}], {n, 40}]

a(n) = Product_{k=1..n} (2n - k)^(c(k)*c(2n - k)), where c is the prime characteristic (A010051).
a(n) = Product_{p+q = 2n, p<=q, and p,q prime} q.
a(n) = A337568(n) / A362641(n).

A362641 Product of the smaller primes, p, in the Goldbach partitions of 2n such that p + q = 2n, p <= q, and p,q prime (or 1 if no Goldbach partition of 2n exists).

Original entry on oeis.org

1, 2, 3, 3, 15, 5, 21, 15, 35, 21, 165, 385, 273, 55, 1001, 39, 2805, 7735, 133, 561, 13585, 273, 5865, 124355, 5187, 1265, 391391, 741, 27115, 19605131, 1767, 64515, 5766215, 217, 374187, 12212915, 313131, 170085, 142635185, 63973, 902451, 13147103255, 223041, 101065, 818183948197

Offset: 1

Wesley Ivan Hurt, Apr 28 2023

			a(10) = 21; 2*10 = 20 has two Goldbach partitions, namely 17+3 and 13+7. The product of the smaller parts of these partitions, is 3*7 = 21.

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A010051, A045917, A337568 (product of all prime parts), A362640 (product of the larger primes q).

Table[Product[k^((PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1])), {k, n}], {n, 40}]

a(n) = Product_{k=1..n} k^(c(k)*c(2n - k)), where c is the prime characteristic (A010051).
a(n) = Product_{p+q = 2n, p<=q, and p,q prime} p.
a(n) = A337568(n) / A362640(n).

cp's OEIS Frontend

A226237 Sum of the parts in the Goldbach partitions of 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A228553 Sum of the products formed by multiplying together the smaller and larger parts of each Goldbach partition of 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A238134 Number of primes p < n with q = floor((n-p)/4) and prime(q) - q + 1 both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A238701 Number of primes p < n with q = floor((n-p)/4) and q^2 - 2 both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A258713 A001172(n)/2: Least k such that 2k is a sum of two odd primes in exactly n ways.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A279481 Count the primes appearing in each interval [p,q] where (p,q) is a Goldbach partition of 2n, and then add the results.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A281687 Number of partitions of 2*n into the sum of two totient numbers (A002202).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A325142 a(n) = k if (n - k, n + k) is the centered Goldbach partition of 2n if it exists and -1 otherwise.

Views

Author