A071917 -id:A071917 - cp's OEIS frontend

A282516 Number T(n,k) of k-element subsets of [n] having a prime element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 2, 4, 1, 0, 0, 3, 5, 2, 2, 0, 0, 3, 7, 6, 4, 2, 0, 0, 4, 9, 10, 11, 7, 1, 0, 0, 4, 11, 18, 21, 13, 7, 2, 0, 0, 4, 14, 26, 34, 31, 20, 7, 3, 0, 0, 4, 18, 37, 53, 59, 51, 32, 11, 2, 0, 0, 5, 21, 47, 82, 110, 117, 85, 35, 12, 2, 0

Offset: 0

Alois P. Heinz, Feb 17 2017

			Triangle T(n,k) begins:
  0;
  0, 0;
  0, 1,  1;
  0, 2,  2,  0;
  0, 2,  4,  1,  0;
  0, 3,  5,  2,  2,   0;
  0, 3,  7,  6,  4,   2,   0;
  0, 4,  9, 10, 11,   7,   1,  0;
  0, 4, 11, 18, 21,  13,   7,  2,  0;
  0, 4, 14, 26, 34,  31,  20,  7,  3,  0;
  0, 4, 18, 37, 53,  59,  51, 32, 11,  2, 0;
  0, 5, 21, 47, 82, 110, 117, 85, 35, 12, 2, 0;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000004, A000720, A071917, A320678, A320679, A320680, A320681, A320682, A320683, A320684, A320685.
Row sums give A127542.
Main diagonal gives A185012.
First lower diagonal gives A282518.
T(2n,n) gives A282517.

b:= proc(n, s) option remember; expand(`if`(n=0,
      `if`(isprime(s), 1, 0), b(n-1, s)+x*b(n-1, s+n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..16);

b[n_, s_] := b[n, s] = Expand[If[n==0, If[PrimeQ[s], 1, 0], b[n-1, s] + x*b[n-1, s+n]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

A292918 Let A_n be a square n X n matrix with entries A_n(i,j)=1 if i+j is prime, and A_n(i,j)=0 otherwise. Then a(n) counts the 1's in A_n.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 19, 23, 29, 37, 43, 51, 57, 63, 71, 81, 89, 97, 105, 113, 123, 135, 145, 157, 169, 181, 195, 209, 221, 235, 249, 263, 277, 293, 309, 327, 345, 363, 381, 401, 419, 439, 457, 475, 495, 515, 533, 551, 571, 591, 613, 637, 659, 683, 709, 735

Offset: 1

Anthony Hernandez, Sep 26 2017

nonn

Bertrand's postulate guarantees for every integer n the existence of at least one prime q with n < q < 2n. Equivalently, A(n) has at least one skew diagonal below the main skew diagonal whose entries will be equal to 1.

			         |1 1 0 1 0|
         |1 0 1 0 1|
   A_5 = |0 1 0 1 0| and so a(5) = 11.
         |1 0 1 0 0|
         |0 1 0 0 0|

David A. Corneth, Table of n, a(n) for n = 1..10000
William Dowling and Nadia Lafreniere, Homomesy on permutations with toggling actions, arXiv:2312.02383 [math.CO], 2023. See page 10.

Cf. A000040, A000720, A069879, A071917.

sol:=[]; for n in [1..56] do k:=0; for i,j in [1..n] do if IsPrime(i+j) then k:=k+1; end if; end for; Append(~sol,k);end for; sol; // Marius A. Burtea, Aug 29 2019

with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1,
      a(n-1)+2*(pi(2*n-1)-pi(n)))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 29 2017

A[n_] := Table[Boole[PrimeQ[i + j]], {i, 1, n}, {j, 1, n}]; a[n_] := Count[Flatten[A[n]], 1];
(* or, after Alois P. Heinz (200 times faster): *)
a[1] = 1; a[n_] := a[n] = a[n-1] + 2(PrimePi[2n-1] - PrimePi[n]);
Array[a, 80] (* Jean-François Alcover, Sep 29 2017 *)

first(n) = {my(res = vector(n), pn = 0, p2n1 = 1); res[1] = 1; for(i = 2, n,
if(isprime(i), pn++); if(isprime(2*i-1), p2n1++); res[i] = res[i-1] + 2*(p2n1 - pn)); res} \\ David A. Corneth, Aug 31 2019

from sympy import primepi
from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n==1 else a(n - 1) + 2*(primepi(2*n - 1) - primepi(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Dec 13 2017, after Alois P. Heinz

From Alois P. Heinz, Sep 29 2017: (Start)
a(n) = a(n-1) + 2 * (pi(2*n-1) - pi(n)) for n > 1, a(1) = 1.
a(n) = A069879(n) + 1 = 2*A071917(n) + 1. (End)
a(n) = Sum_{i=1..n} (pi(n+i) - pi(i)), where pi = A000720. - Ridouane Oudra, Aug 29 2019
a(n) = Sum_{p <= 2n+1, p prime} min(p-1, 2n+1-p). - Ridouane Oudra, Oct 30 2023

A069879 Number of pairs {i,j} with i different from j; 1<=i<=n; 1<= j <=n such that i+j is a prime number.

Original entry on oeis.org

0, 2, 4, 8, 10, 14, 18, 22, 28, 36, 42, 50, 56, 62, 70, 80, 88, 96, 104, 112, 122, 134, 144, 156, 168, 180, 194, 208, 220, 234, 248, 262, 276, 292, 308, 326, 344, 362, 380, 400, 418, 438, 456, 474, 494, 514, 532, 550, 570, 590, 612, 636, 658, 682, 708, 734

Offset: 1

Santi Spadaro, May 04 2002

nonn

Partial sums of 2*A060715(n).

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 0,
      b(n-1)+pi(2*n-1)-pi(n))
    end:
a:= n-> 2*b(n):
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 29 2017

Table[2*Count[Subsets[Range[n],{2}],?(PrimeQ[Total[#]]&)],{n,50}] (* _Harvey P. Dale, Jan 23 2015 *)

a(n) = 2 * A071917(n). - Alois P. Heinz, Sep 29 2017

A140199 a(n) = the number of pairs of (not necessarily distinct) positive integers j and k where j <= n and k <= n such that k+j is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 12, 15, 19, 22, 26, 29, 32, 36, 41, 45, 49, 53, 57, 62, 68, 73, 79, 85, 91, 98, 105, 111, 118, 125, 132, 139, 147, 155, 164, 173, 182, 191, 201, 210, 220, 229, 238, 248, 258, 267, 276, 286, 296, 307, 319, 330, 342, 355, 368, 382, 396, 409, 422

Offset: 1

Leroy Quet, Jun 15 2008

nonn

			For n = 4 there are 5 pairs of positive integers, each <= n=4, that sum to a prime: 1+1=2, 1+2=3, 2+3=5, 1+4=5 and 3+4=7.

Cf. A108954, A071917.

A108954 := proc(n) numtheory[pi](2*n)-numtheory[pi](n) ; end: A140199 := proc(n) add( A108954(k),k=1..n) ; end: seq(A140199(n),n=1..80) ; # R. J. Mathar, Jun 19 2008

Table[Count[Union[Sort/@Tuples[Range[n],{2}]],?(PrimeQ[Total[#]]&)],{n,60}] (* _Harvey P. Dale, Mar 21 2021 *)

a(n) = sum{k=1 to n} A108954(k). (A108954(n) = pi(2n)-pi(n), where pi(n) is the number of primes that are <= n). a(n) = A071917(n)+1.

More terms from R. J. Mathar, Jun 19 2008

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A282516 Number T(n,k) of k-element subsets of [n] having a prime element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A292918 Let A_n be a square n X n matrix with entries A_n(i,j)=1 if i+j is prime, and A_n(i,j)=0 otherwise. Then a(n) counts the 1's in A_n.

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A069879 Number of pairs {i,j} with i different from j; 1<=i<=n; 1<= j <=n such that i+j is a prime number.

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A140199 a(n) = the number of pairs of (not necessarily distinct) positive integers j and k where j <= n and k <= n such that k+j is prime.

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