A037020 Numbers whose sum of proper (or aliquot) divisors is a prime.

4, 8, 21, 27, 32, 35, 39, 50, 55, 57, 63, 65, 77, 85, 98, 111, 115, 125, 128, 129, 155, 161, 171, 175, 185, 187, 189, 201, 203, 205, 209, 221, 235, 237, 242, 245, 265, 275, 279, 291, 299, 305, 309, 319, 323, 324, 325, 327, 335, 338, 341, 365, 371, 377, 381

Offset: 1

Author

Felice Russo, Dec 11 1999

Keywords

• nonn
• easy
• nice

Comments

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

Examples

			The aliquot divisors of 27 are 1, 3, and 9, whose sum is 13, a prime, so 27 is a term.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Paul Pollack, Some arithmetic properties of the sum of proper divisors and the sum of prime divisors, Illinois J. Math. 58:1 (2014), pp. 125-147.

Crossrefs

Cf. A001065, A053868, A053869, A010051.

Programs

Haskell

a037020 n = a037020_list !! (n-1)
a037020_list = filter ((== 1) . a010051' . a001065) [1..]
-- Reinhard Zumkeller, Nov 01 2015, Sep 15 2011

Magma

[n: n in [2..500] | IsPrime(SumOfDivisors(n)-n)]; // Vincenzo Librandi, Nov 01 2016

Mathematica

Select[Range[400],PrimeQ[DivisorSigma[1,#]-#]&] (* Harvey P. Dale, May 09 2011 *)

PARI

isok(n) = isprime(sigma(n) - n); \\ Michel Marcus, Nov 01 2016

Formula

A001065(a(n)) is in A000040.

Pollack proves that a(n) >> n log n. - Charles R Greathouse IV, Jun 28 2021