A274876 The number of ways (2n)^2 is expressible as (p+1)(q+1) where p and q are distinct primes.

0, 0, 1, 0, 0, 3, 0, 1, 2, 0, 0, 4, 0, 1, 1, 1, 0, 4, 0, 2, 3, 1, 0, 4, 0, 1, 1, 2, 0, 5, 0, 1, 2, 1, 1, 4, 0, 1, 3, 1, 0, 7, 0, 2, 4, 0, 0, 4, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0, 0, 9, 0, 0, 2, 0, 0, 5, 0, 3, 0, 2, 0, 8, 0, 0, 2, 2, 2, 6, 0, 2, 2, 1, 0, 6, 0, 1, 1, 2, 0, 8, 1, 1, 1, 0, 0, 2, 0, 1, 0, 4, 0, 4, 0, 2, 5

Offset

1,6

Comments

No odd number squared is expressible as (p+1)(q+1) where p and q are distinct primes, since q must be odd and therefore (q+1) is even.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

a(1) = 0 since 2 is not expressible as (p+1)(q+1); same for a(2); a(3) = 1 since 6^2 = 36 = (2+1)(11+1); a(6) = 3 since 12^2 = 144 = (2+1)(47+1) = (5+1)(23+1) = (7+1)(17+1); a(9) = 2 since 18^2 = 324 = (2+1)(107+1) = (5+1)(53+1); etc.

Mathematica

f[n_] := Block[{c = 0, p = 2}, While[p < 2n -1, If[ PrimeQ[(2n)^2/(p +1) -1], c++]; p = NextPrime@ p]; c]; Array[f, 105]

Prog

(PARI) a(n)=sumdiv(4*n^2, d, d<2*n && isprime(d-1) && isprime(4*n^2/d-1)) \\ Charles R Greathouse IV, Jul 10 2016

Crossrefs

Cf. A274848, A274877.

Sequence in context: A303301 A160499 A329272 * A065718 A025428 A199176

Adjacent sequences: A274873 A274874 A274875 * A274877 A274878 A274879

Keyword

nonn

Author

Zak Seidov and Robert G. Wilson v, Jul 10 2016

Status

approved