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%I #41 Feb 21 2022 01:00:32
%S 4,8,21,27,32,35,39,50,55,57,63,65,77,85,98,111,115,125,128,129,155,
%T 161,171,175,185,187,189,201,203,205,209,221,235,237,242,245,265,275,
%U 279,291,299,305,309,319,323,324,325,327,335,338,341,365,371,377,381
%N Numbers whose sum of proper (or aliquot) divisors is a prime.
%C Assuming the Goldbach conjecture, it is easy to show that all primes, except 2 and 5, are the sum of the proper divisors of some number. - _T. D. Noe_, Nov 29 2006
%H T. D. Noe, <a href="/A037020/b037020.txt">Table of n, a(n) for n = 1..10000</a>
%H Paul Pollack, <a href="https://doi.org/10.1215/ijm/1427897171">Some arithmetic properties of the sum of proper divisors and the sum of prime divisors</a>, Illinois J. Math. 58:1 (2014), pp. 125-147.
%F A001065(a(n)) is in A000040.
%F Pollack proves that a(n) >> n log n. - _Charles R Greathouse IV_, Jun 28 2021
%e The aliquot divisors of 27 are 1, 3, and 9, whose sum is 13, a prime, so 27 is a term.
%t Select[Range[400],PrimeQ[DivisorSigma[1,#]-#]&] (* _Harvey P. Dale_, May 09 2011 *)
%o (Haskell)
%o a037020 n = a037020_list !! (n-1)
%o a037020_list = filter ((== 1) . a010051' . a001065) [1..]
%o -- _Reinhard Zumkeller_, Nov 01 2015, Sep 15 2011
%o (PARI) isok(n) = isprime(sigma(n) - n); \\ _Michel Marcus_, Nov 01 2016
%o (Magma) [n: n in [2..500] | IsPrime(SumOfDivisors(n)-n)]; // _Vincenzo Librandi_, Nov 01 2016
%Y Cf. A001065, A053868, A053869, A010051.
%K nonn,easy,nice
%O 1,1
%A _Felice Russo_, Dec 11 1999
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