A238570 a(n) = |{0 < k < n: pi((k+1)^2) - pi(k^2) and pi(n^2) - pi(k^2) are both prime}|, where pi(x) denotes the number of primes not exceeding x.
0, 1, 1, 1, 3, 4, 2, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 2, 2, 2, 4, 5, 2, 5, 3, 6, 4, 5, 4, 1, 2, 2, 6, 4, 2, 1, 3, 1, 1, 5, 5, 1, 6, 3, 3, 7, 4, 6, 1, 4, 5, 3, 4, 4, 7, 6, 4, 7, 6, 6, 1, 3, 3, 5, 6, 6, 3, 4, 9, 6, 4, 2, 5, 3, 8, 3, 3, 6, 8, 6
Offset: 1
Keywords
Examples
a(8) = 1 since pi(8^2) - pi(7^2) = 18 - 15 = 3 is prime. a(61) = 1 since pi(27^2) - pi(26^2) = 129 - 122 = 7 and pi(61^2) - pi(26^2) = 519 - 122 = 397 are both prime. a(86) = 1 since pi(3^2) - pi(2^2) = 4 - 2 = 2 and pi(86^2) - pi(2^2) = 939 - 2 = 937 are both prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
- Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.
Programs
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Mathematica
p[k_,n_]:=PrimeQ[PrimePi[(k+1)^2]-PrimePi[k^2]]&&PrimeQ[PrimePi[n^2]-PrimePi[k^2]] a[n_]:=Sum[If[p[k,n],1,0],{k,1,n-1}] Table[a[n],{n,1,80}]
Comments