A274877 Least number, m, such that m^2 is expressible in just n ways as (p+1)(q+1) where p and q are distinct primes.

1, 6, 18, 12, 24, 60, 156, 84, 144, 120, 816, 336, 360, 1224, 840, 924, 2184, 1800, 2640, 7200, 1260, 3960, 7140, 8400, 3780, 5040, 2520, 9360, 12600, 20160, 11340, 10080, 15120, 19656, 16380, 41580, 18480, 48720, 34320, 25200, 54600, 27720, 87360, 134640, 60060, 73920, 32760, 43680, 159600, 143640, 55440, 85800, 96096, 65520, 131040, 120120, 157080, 154440, 98280, 191520, 166320

Offset

0,2

Comments

Records: 1, 6, 12, 24, 60, 84, 120, 336, 360, 840, ..., .

From David A. Corneth, Jul 10 2025: (Start)

a(n) is even for n > 0. If a(n) is odd and n > 0 then there are two distinct even primes.

For terms t where 1 < t <= 10^8 we have 6 | t. Is any term > 1 divisible by 6? (End)

Links

David A. Corneth, Table of n, a(n) for n = 0..275

David A. Corneth, PARI program

Example

a(0) = 1 since 1 is not expressible as (p+1)(q+1), in fact no odd number is expressible this way;
a(1) = 6 since 6^2 = 36 = (2+1)(11+1);
a(2) = 18 since 18^2 = 324 = (2+1)(107+1) = (5+1)(53+1);
a(3) = 12 since 12^2 = 144 = (2+1)(47+1) = (5+1)(23+1) = (7+1)(17+1); etc.

Mathematica

(* first compute A274876: f[n_] := f[n] = Block[{c = 0, p = 2}, While[p < 2n -1, If[ PrimeQ[(2n)^2/(p +1) -1], c++]; p = NextPrime@ p]; c]; f[0] = 1; then *) Table[(2Select[ Range@ 5000, f@# == n &])[[1]], {n, 0, 10}]

Prog

(PARI) \\ See Corneth link

Crossrefs

Cf. A274848, A274877.

Sequence in context: A248461 A129870 A331056 * A091014 A097370 A252897

Adjacent sequences: A274874 A274875 A274876 * A274878 A274879 A274880

Keyword

nonn

Author

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

Status

approved