A002375 -id:A002375 - cp's OEIS frontend

A261627 Number of primes p such that n-(pn'-1) and n+(pn'-1) are both prime, where n' is 1 or 2 according as n is odd or even.

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

Offset: 1

Zhi-Wei Sun, Aug 27 2015

nonn

Conjecture: a(n) > 0 for all n > 6, and a(n) = 1 only for n = 5, 7, 10, 11, 12, 19, 22, 30, 34, 44, 46, 72, 142.
This is stronger than Goldbach's conjecture (A002375) and Lemoine's conjecture (A046927).
I have verified the conjecture for n up to 10^8.
Verified for n up to 10^9. - Mauro Fiorentini, Jul 05 2023
Conjecture verified for n < 1.2 * 10^12. - Jud McCranie, Aug 26 2023

			a(19) = 1 since 13, 19-(13-1) = 7 and 19+(13-1) = 31 are all prime.
a(142) = 1 since 41, 142-(2*41-1) = 61 and 142+(2*41-1) = 223 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2015.

Cf. A000040, A002372, A002375, A046927, A219055, A237284, A261628.

Do[r=0;Do[If[PrimeQ[n-(3+(-1)^n)/2*Prime[k]+1]&&PrimeQ[n+(3+(-1)^n)/2*Prime[k]-1],r=r+1],{k,1,PrimePi[2n/(3+(-1)^n)]}];Print[n," ",r];Continue,{n,1,80}]

A274987 Primes p such that A274601(p) is a prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 59, 61, 73, 79, 83, 89, 101, 103, 109, 127, 137, 139, 149, 173, 179, 193, 223, 229, 257, 263, 293, 307, 313, 337, 347, 349, 359, 367, 389, 397, 409, 419, 439, 449, 461, 467, 487, 491, 521, 547, 571, 577, 599, 601, 619, 631

Offset: 1

Lei Zhou, Nov 11 2016

It is conjectured that the sequence is infinite.

This sequence is also the list of primes with k trits that are used in decomposition of 2*3^k into the sum of such two primes. k>=1.

			For p=2, A274601(p) = 4, which is not a prime, so ignore 2.
For p=3, A274601(p) = 3, which is a prime, so a(1)=3.
For p=5, A274601(p) = 13, which is a prime, so a(2)=5.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A134021, A274601, A002375.

p = 2; Table[While[p = NextPrime[p]; cp = 2*3^(Floor[Log[3, 2*p - 1]]) - p; !PrimeQ[cp]]; p, {n, 1, 56}]

A137791 Number of ways to write n as sum of two positive numbers having no prime gaps in their factorization.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Feb 11 2008

nonn

			a(20) = #{1+19,2+18,3+17,4+16,5+15,7+13,8+12,9+11} = 8;
a(21) = #{2+19,3+18,4+17,5+16,6+15,8+13,9+12} = 7;
a(22) = #{3+19,4+18,5+17,6+16,7+15,9+13,11+11} = 7;
a(23) = #{4+19,5+18,6+17,7+16,8+15,11+12} = 6;
a(24) = #{1+23,5+19,6+18,7+17,8+16,9+15,11+13,12+12} = 8.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A073491, A002375, A061358, A071330, A137792, A137793.

A202472 Goldbach's Problem extended to subtraction: number of decompositions of 2n into unordered differences of two primes, p, q, where p < 2n < q.

Original entry on oeis.org

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

Offset: 1

James D. Klein, Dec 19 2011

nonn

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

Extension of A002375.

Bisection of A092953.

Table[Length[Select[Prime[Range[PrimePi[2*n]]], PrimeQ[2*n + #] &]], {n, 100}] (* T. D. Noe, Apr 16 2013 *)

a(n)=my(s);forprime(p=2,2*n,s+=isprime(2*n+p));s \\ Charles R Greathouse IV, Dec 19 2011
(C++)
#include 
using namespace std;
int main()
{ int p[25] = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97};
  int count, istart = 2;
  for(int n=1; n<=25; n++)
  {
      if(2*n>p[istart]) istart++;
      count = 0;
      for(int j=1; p[j]<2*n; j++)
        for(int i=istart; p[i]-p[j]<=2*n; i++)
          if(p[i]-p[j]==2*n) count++;
      cout << n << ". " << count << endl;
  }
    return 0;
} // code for the first 25 integers, James D. Klein, Dec 21 2011

a(n) = A092953(2*n). - Bill McEachen, May 24 2024

A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.

Original entry on oeis.org

6, 8, 8, 10, 12, 14, 18, 24, 32, 48, 72, 110, 174, 274, 438, 704, 1134, 1830, 2952, 4762, 7698, 12450, 20128, 32560, 52660, 85168, 137752, 222844, 360564, 583392, 943902, 1527222, 2471074, 3998274, 6469334, 10467566, 16936850, 27404300, 44341050, 71745324

Offset: 1

Vladimir Shevelev, Mar 16 2013

nonn

Conjecture. lim a(n+1)/a(n)=phi as n goes to infinity (phi=golden ratio).

			Let n=6. Since a(4) = 10, a(5) = 12 and g(10) = g(12) = 7, then a(6) = 7 + 7 = 14.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000045, A002375, A025019, A216835.

a[1] = 6; a[2] = 8; g[n_] := Module[{tmp,k=1}, While[!PrimeQ[n-(tmp=NextPrime[n,-k])], k++]; tmp]; a[n_] := a[n] = g[a[n-1]] + g[a[n-2]]; Table[a[n], {n,1,100}]

For n>=5, a(n) = A216835(n-3) + A216835(n-4).

A216835 Fibonacci + Goldbach (dual sequence to A216275). a(1)=5, a(2)=7 and for n>=3, a(n) = g(a(n-1) + a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.

Original entry on oeis.org

5, 7, 7, 11, 13, 19, 29, 43, 67, 107, 167, 271, 433, 701, 1129, 1823, 2939, 4759, 7691, 12437, 20123, 32537, 52631, 85121, 137723, 222841, 360551, 583351, 943871, 1527203, 2471071, 3998263, 6469303, 10467547, 16936753, 27404297, 44341027, 71745313, 116086303

Offset: 1

Vladimir Shevelev, Mar 16 2013

nonn

Conjecture. lim a(n+1)/a(n)=phi as n goes to infinity (phi=golden ratio).

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000045, A002375, A025019, A216275.

a[1] = 5; a[2] = 7; g[n_] := Module[{tmp,k=1}, While[!PrimeQ[n-(tmp=NextPrime[n,-k])], k++]; tmp]; a[n_] := a[n] = g[a[n-1] + a[n-2]]; Table[a[n], {n,1,100}]

a(n) = g(A216275(n+2)).

A230227 Primes p with 3*p - 10 also prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 47, 53, 59, 61, 67, 79, 83, 89, 97, 101, 107, 109, 131, 137, 151, 157, 163, 167, 173, 191, 193, 199, 223, 229, 251, 257, 269, 277, 283, 307, 313, 317, 331, 347, 353, 367, 373, 397, 401, 409

Offset: 1

Zhi-Wei Sun, Oct 12 2013

nonn

Conjecture: For any integer n > 4 not equal to 76, we have 2*n = p + q for some terms p and q from the sequence.

This is stronger than Goldbach's conjecture for even numbers.

			a(1) = 5 since 3*5 - 10 = 5 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.

Cf. A000040, A002375, A230138, A230140, A230141, A230217, A230219, A230223, A230224, A230230.

PQ[p_]:=PQ[p]=p>3&&PrimeQ[3p-10]
m=0
Do[If[PQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,80}]
Select[Prime[Range[100]],PrimeQ[3#-10]&] (* Harvey P. Dale, Jun 28 2015 *)

A230254 Number of ways to write n = p + q with p and (p+1)*q/2 + 1 both prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 14 2013

nonn

Conjecture: a(n) > 0 for all n > 3.
We have verified this for n up to 10^8.
We also have some similar conjectures, for example, any integer n > 3 not equal to 17 or 66 can be written as p + q with p and (p+1)*q/2 - 1 both prime.

			a(15) = 1 since 15 = 5 + 10 with 5 and (5+1)*10/2+1 = 31 both prime.
a(30) = 1 since 30 = 2 + 28 with 2 and (2+1)*28/2+1 = 43 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.

Cf. A000040, A002375, A219864, A230252, A227908, A227909, A230241.

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

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

A306513 The number of unordered pairs of coprime integers q and r such that phi(q) + phi(r) = 2n.

Original entry on oeis.org

1, 1, 5, 7, 12, 10, 19, 18, 20, 21, 35, 32, 39, 42, 38, 37, 48, 46, 45, 58, 64, 63, 69, 73, 58, 93, 71, 70, 81, 92, 72, 113, 96, 94, 90, 100, 79, 158, 120, 95, 131, 153, 84, 147, 129, 132, 126, 150, 92, 179, 157, 150, 149, 187, 92, 224, 177, 166, 173, 207, 124

Offset: 1

Robert G. Wilson v, Feb 20 2019

nonn

Paul Erdős and Leo Moser conjectured that, for any even number 2n, there exist integers q and r such that phi(q) + phi(r) = 2n with gcd(q, r) = 1. Adding to this conjecture the requirement that q and r be prime yields the Goldbach Conjecture. The replacement of the requirement that q and r be prime with the relaxed requirement that they be coprime was done in an effort to solve the Goldbach Conjecture.

			a(1) = 1 with {q, r} = {1,2};
a(2) = 1 with {q, r} = {3,4};
a(3) = 5 because phi(q) + phi(r) = 6 for the pairs {q, r} = {3,5}, {3,8}, {3,10}, {4,5} & {5,6}; etc.

George E. Andrews, Number Theory, Chapter 6, Arithmetic Functions, Section 6-1, Combinatorial Study of Phi(n), pp. 75-82, Dover Publishing, NY, 1971.

Robert G. Wilson v, Table of n, a(n) for n = 1..1150
Eric W. Weisstein's World of Mathematics, Goldbach Conjecture.
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A000010, A005277, A079695, A002375.

f[n_] := Block[{c = 0, q = 1}, While[q < 12n, epq = EulerPhi[q]; r = 12n + 125; While[r > q, If[ GCD[q, r] == 1 && epq + EulerPhi[r] == 2 n, c++]; r--]; q++]; c]; Array[f, 61]

cp's OEIS Frontend

A261627 Number of primes p such that n-(p*n'-1) and n+(p*n'-1) are both prime, where n' is 1 or 2 according as n is odd or even.

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A274987 Primes p such that A274601(p) is a prime.

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A137791 Number of ways to write n as sum of two positive numbers having no prime gaps in their factorization.

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A202472 Goldbach's Problem extended to subtraction: number of decompositions of 2n into unordered differences of two primes, p, q, where p < 2n < q.

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PARI

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A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2*m) is the maximal prime p < 2*m such that 2*m - p is prime.

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A216835 Fibonacci + Goldbach (dual sequence to A216275). a(1)=5, a(2)=7 and for n>=3, a(n) = g(a(n-1) + a(n-2)), where for m>=3, g(2*m) is the maximal prime p < 2*m such that 2*m - p is prime.

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Formula

A230227 Primes p with 3*p - 10 also prime.

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A230254 Number of ways to write n = p + q with p and (p+1)*q/2 + 1 both prime.

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Mathematica

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

A261627 Number of primes p such that n-(pn'-1) and n+(pn'-1) are both prime, where n' is 1 or 2 according as n is odd or even.

A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.

A216835 Fibonacci + Goldbach (dual sequence to A216275). a(1)=5, a(2)=7 and for n>=3, a(n) = g(a(n-1) + a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.