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A069548
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Primes of the form sum 6d/(2 + mu(d)) for some k and all d dividing k.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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