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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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]
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LINKS
| Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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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]
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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.
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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] (* From Alonso del Arte, Feb 17 2011 *)
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PROG
| (PARI) isA069548(n)=isprime(sumdiv(n, d, 6*d/(2+moebius(d))) \\ Charles R Greathouse IV, Feb 17 2011
(PARI) list(lim)=my(v=List(), t); for(n=1, sqrt(lim), if(isA069548(n^2), listput(v, n^2))); for(n=1, sqrt(lim\2), if(isA069548(2*n^2), listput(v, 2*n^2))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Feb 18 2011
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CROSSREFS
| Sequence in context: A139839 A118104 A107211 * A065044 A142313 A153213
Adjacent sequences: A069545 A069546 A069547 * A069549 A069550 A069551
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 18, 2002
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