A088006 Abundance values of numbers whose abundance is (+-1) times a prime.

-2, -5, -2, 3, 2, -19, 19, -41, -7, -41, 17, 2, -109, -2, 7, -47, -271, 199, -127, 71, 2, -37, 499, 2, -71, 199, 353, -811, 1021, -929, 59, -457, -449, -163, -683, -157, 41, -751, 251, 2, -2161, -19, 401, 467, -61, 967, -631, -3659, -2777, 3391, -4421, 269, -2333, -1201, -4969, 4999, -1103, -2647, -1097, 269

Offset: 1

Author

Labos Elemer, Oct 18 2003

Keywords

• sign

Examples

			For n = 4: A088005(4) = 18, sigma(18) = 18 + 9 + 6 + 3 + 2 + 1 = 39, 2*18 = 36, abundance = 39 - 36 = 3 = a(4).
For n = 6: A088005(6) = 25, sigma(25)_= 25 + 5 + 1 = 31, 2*25 = 50, abundance = 31 - 50 = -19 = a(6).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000203, A033880, A077374, A087998, A088005.

Programs

Mathematica

ab[x_] := DivisorSigma[1, x]-2*x Do[If[PrimeQ[s=ab[n]], Print[s]], {n, 1, 10000}]
Select[Table[DivisorSigma[1,n]-2n,{n,7000}],PrimeQ[Abs[#]]&] (* Harvey P. Dale, Aug 21 2011 *)

PARI

list(lim) = {my(ab); for(k = 1, lim, ab = sigma(k) - 2*k; if(isprime(abs(ab)), print1(ab, ", ")));} \\ Amiram Eldar, Feb 16 2025

Formula

Solutions x to sigma(k) - 2k = x where abs(x) is a prime number.

a(n) = A033880(A088005(n)). - Amiram Eldar, Feb 16 2025