A292918 - OEIS

%I #50 Dec 09 2023 09:11:39

%S 1,3,5,9,11,15,19,23,29,37,43,51,57,63,71,81,89,97,105,113,123,135,

%T 145,157,169,181,195,209,221,235,249,263,277,293,309,327,345,363,381,

%U 401,419,439,457,475,495,515,533,551,571,591,613,637,659,683,709,735

%N 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 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 David A. Corneth, <a href="/A292918/b292918.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 From _Alois P. Heinz_, Sep 29 2017: (Start)

%F a(n) = a(n-1) + 2 * (pi(2*n-1) - pi(n)) for n > 1, a(1) = 1.

%F a(n) = A069879(n) + 1 = 2*A071917(n) + 1. (End)

%F a(n) = Sum_{i=1..n} (pi(n+i) - pi(i)), where pi = A000720. - _Ridouane Oudra_, Aug 29 2019

%F a(n) = Sum_{p <= 2n+1, p prime} min(p-1, 2n+1-p). - _Ridouane Oudra_, Oct 30 2023

%e          |1 1 0 1 0|

%e          |1 0 1 0 1|

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

%e          |1 0 1 0 0|

%e          |0 1 0 0 0|

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=1, 1,

%p       a(n-1)+2*(pi(2*n-1)-pi(n)))

%p     end:

%p seq(a(n), n=1..80);  # _Alois P. Heinz_, Sep 29 2017

%t A[n_] := Table[Boole[PrimeQ[i + j]], {i, 1, n}, {j, 1, n}]; a[n_] := Count[Flatten[A[n]], 1];

%t (* or, after _Alois P. Heinz_ (200 times faster): *)

%t a[1] = 1; a[n_] := a[n] = a[n-1] + 2(PrimePi[2n-1] - PrimePi[n]);

%t Array[a, 80] (* _Jean-François Alcover_, Sep 29 2017 *)

%o (Python)

%o from sympy import primepi

%o from sympy.core.cache import cacheit

%o @cacheit

%o def a(n): return 1 if n==1 else a(n - 1) + 2*(primepi(2*n - 1) - primepi(n))

%o print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Dec 13 2017, after _Alois P. Heinz_

%o (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 (PARI) first(n) = {my(res = vector(n), pn = 0, p2n1 = 1); res[1] = 1; for(i = 2, n,

%o 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 Cf. A000040, A000720, A069879, A071917.

%K nonn

%O 1,2

%A _Anthony Hernandez_, Sep 26 2017