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%I A292918 #50 Dec 09 2023 09:11:39
%S A292918 1,3,5,9,11,15,19,23,29,37,43,51,57,63,71,81,89,97,105,113,123,135,
%T A292918 145,157,169,181,195,209,221,235,249,263,277,293,309,327,345,363,381,
%U A292918 401,419,439,457,475,495,515,533,551,571,591,613,637,659,683,709,735
%N 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.
%C A292918 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.
%H A292918 David A. Corneth, <a href="/A292918/b292918.txt">Table of n, a(n) for n = 1..10000</a>
%H A292918 William Dowling and Nadia Lafreniere, <a href="https://arxiv.org/abs/2312.02383">Homomesy on permutations with toggling actions</a>, arXiv:2312.02383 [math.CO], 2023. See page 10.
%F A292918 From _Alois P. Heinz_, Sep 29 2017: (Start)
%F A292918 a(n) = a(n-1) + 2 * (pi(2*n-1) - pi(n)) for n > 1, a(1) = 1.
%F A292918 a(n) = A069879(n) + 1 = 2*A071917(n) + 1. (End)
%F A292918 a(n) = Sum_{i=1..n} (pi(n+i) - pi(i)), where pi = A000720. - _Ridouane Oudra_, Aug 29 2019
%F A292918 a(n) = Sum_{p <= 2n+1, p prime} min(p-1, 2n+1-p). - _Ridouane Oudra_, Oct 30 2023
%e A292918          |1 1 0 1 0|
%e A292918          |1 0 1 0 1|
%e A292918    A_5 = |0 1 0 1 0| and so a(5) = 11.
%e A292918          |1 0 1 0 0|
%e A292918          |0 1 0 0 0|
%p A292918 with(numtheory):
%p A292918 a:= proc(n) option remember; `if`(n=1, 1,
%p A292918       a(n-1)+2*(pi(2*n-1)-pi(n)))
%p A292918     end:
%p A292918 seq(a(n), n=1..80);  # _Alois P. Heinz_, Sep 29 2017
%t A292918 A[n_] := Table[Boole[PrimeQ[i + j]], {i, 1, n}, {j, 1, n}]; a[n_] := Count[Flatten[A[n]], 1];
%t A292918 (* or, after _Alois P. Heinz_ (200 times faster): *)
%t A292918 a[1] = 1; a[n_] := a[n] = a[n-1] + 2(PrimePi[2n-1] - PrimePi[n]);
%t A292918 Array[a, 80] (* _Jean-François Alcover_, Sep 29 2017 *)
%o A292918 (Python)
%o A292918 from sympy import primepi
%o A292918 from sympy.core.cache import cacheit
%o A292918 @cacheit
%o A292918 def a(n): return 1 if n==1 else a(n - 1) + 2*(primepi(2*n - 1) - primepi(n))
%o A292918 print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Dec 13 2017, after _Alois P. Heinz_
%o A292918 (Magma) 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
%o A292918 (PARI) first(n) = {my(res = vector(n), pn = 0, p2n1 = 1); res[1] = 1; for(i = 2, n,
%o A292918 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
%Y A292918 Cf. A000040, A000720, A069879, A071917.
%K A292918 nonn
%O A292918 1,2
%A A292918 _Anthony Hernandez_, Sep 26 2017

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