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.
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
Keywords
Examples
|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|
Links
- 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.
Programs
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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
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Maple
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 -
Mathematica
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 *) -
PARI
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 -
Python
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
Formula
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) = 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
Comments