A069548 Primes of the form sum 6d/(2 + mu(d)) for some k and all d dividing k.

2, 47, 107, 191, 281, 431, 587, 593, 661, 971, 1097, 1213, 1217, 2357, 2549, 2699, 5807, 5869, 6469, 6911, 7039, 7873, 8423, 8747, 10799, 11261, 11821, 11981, 14867, 15551, 16411, 16427, 18223, 19937, 22877, 22961, 25153, 28573, 29531, 30467

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

Sequence in context: A248862 A118104 A107211 * A065044 A142313 A264776

Adjacent sequences: A069545 A069546 A069547 * A069549 A069550 A069551

Keyword

nonn

Author

Benoit Cloitre, Apr 18, 2002

Status

approved