A282849 Number of divisors k of n such that (n + k^2)/k is a prime.

1, 2, 0, 2, 0, 4, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 8, 0, 0, 0, 2, 0, 4, 0, 0, 0, 4, 0, 8, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 8, 0, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 4, 0, 0, 0, 2, 0, 8, 0, 0, 0, 4, 0, 4, 0, 0, 0, 4, 0, 6

Offset: 1

Author

Juri-Stepan Gerasimov, Feb 24 2017

Keywords

• nonn

Comments

Except for the single case of a(1)=1 all terms are even. - Robert G. Wilson v, Feb 25 2017

First occurrence of 2k: 3, 2, 6, 90, 30, 390, 690, 420, 210, 4290, 3990, 8778, 2310, 3570, 4830, 11550, 38850, 84630, 66990, 79170, 39270, 30030, 51870, 46410, 43890, ..., . - Robert G. Wilson v, Feb 25 2017

Examples

			a(6) = 4 because (6 + 1^2)/1 = 7 is prime, (6 + 2^2)/2 = 5 is prime, (6 + 3^2)/3 = 5 is prime, (6 + 6^2)/6 = 7 is prime, where 1, 2, 3 and 6 are divisors of 6.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Records and indices of records.

Crossrefs

Cf. A088627 (number of divisors k of n such that (n + 2*k^2)/k is prime), A047255.

Programs

Mathematica

f[n_] := Block[{d = Divisors@ n}, Length@ Select[d, PrimeQ[(n + #^2)/#] &]]; Array[f, 105] (* Robert G. Wilson v, Feb 25 2017 *)
Table[DivisorSum[n, 1 &, PrimeQ[(n + #^2)/#] &], {n, 105}] (* Michael De Vlieger, Nov 15 2017 *)

PARI

a(n) = sumdiv(n, k, isprime((n+k^2)/k)); \\ Michel Marcus, Feb 26 2017

Formula

a(1) = 1; for n > 0: a(2n) = 2*A088627(n), a(2n + 1) = 0.