OFFSET
1,1
COMMENTS
mu is the Moebius mu function, see A008683.
The function is the sum of 3d or -d (mu(d) -1 or 1, resp.) over the squarefree divisors d of n, plus 3*sigma(n). As a result, for n > 1, the squarefree part is even, and thus n generates odd numbers only when 3*sigma(n) is odd. This happens only when n is a square or twice a square, as conjectured by Alonso Del Arte. - Charles R Greathouse IV, Feb 18 2011
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
Primes of the form sum_{d|k} 6d/(2 + mu(d)) for some k.
Primes of the form 3*sigma(n) + sum_{d|k, d squarefree} d(6/(2 + mu(d)) - 3) for some k. - Charles R Greathouse IV, Feb 18 2011
EXAMPLE
For k = 9, the divisors d are 1, 3 and 9. We see that 6 * 1 / (2 + mu(1)) = 2, that 6 * 3 / (2 + mu(3)) = 18 and that 6 * 9 / (2 + mu(9)) = 27. Then, 2 + 18 + 27 = 47, which is prime, so it is in the list.
MATHEMATICA
cloitreMu[k_] := Plus@@Table[6Divisors[k][[d]] / (2 + MoebiusMu[Divisors[k][[d]]]), {d, DivisorSigma[0, k]}]; Take[Union[Select[Table[cloitreMu[n], {n, 10^5}], PrimeQ]], 40] (* Alonso del Arte, Feb 17 2011 *)
PROG
(PARI) f(n)=sumdiv(n, d, 6*d/(2+moebius(d)))
list(lim)=my(v=List(), t); for(n=1, sqrtint(lim\3), if(isprime(t=f(n^2)) && t<=lim, listput(v, t))); for(n=1, sqrtint(lim\8), if(isprime(t=f(2*n^2)) && t<=lim, listput(v, t))); Set(v) \\ Charles R Greathouse IV, Feb 18 2011; revised Sep 26 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Apr 18, 2002
STATUS
approved