A096847 Numbers k such that A094471(k) is prime.

3, 4, 8, 36, 100, 128, 324, 400, 1296, 1600, 1936, 2116, 3364, 4356, 10404, 11236, 20736, 22500, 26244, 27556, 28900, 30976, 38416, 40000, 52900, 53824, 57600, 60516, 88804, 93636, 107584, 108900, 115600, 123904, 125316, 129600, 211600, 215296, 220900, 256036

Offset: 1

Author

Labos Elemer, Jul 15 2004

Keywords

• nonn

Comments

Old name was "Solutions to {A094471[x]=prime} that is to {x; x*tau[x]-sigma[x]=prime}."

All terms after the first are even, because A094471(n) is even if n is odd. The first term == 2 (mod 4) is a(135) = 9653618. - Robert Israel, Nov 11 2015

Except for 3, all the terms are either even squares or twice squares. - Amiram Eldar, Feb 14 2025

Examples

			8 is a term since 8*tau(8) - sigma(8) = 8*4 - 15 = 32 - 15 = 17 is a prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Harvey P. Dale)

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A094471, A096848.

Programs

Maple

A094471:= n -> n*numtheory:-tau(n) - numtheory:-sigma(n):
select(t -> isprime(A094471(t)), 2*[3/2,$1..10^6]); # Robert Israel, Nov 11 2015

Mathematica

Do[s=n*DivisorSigma[0, n]-DivisorSigma[1, n]; If[PrimeQ[s], Print[{n, s}]; ta[[u]]=n; tb[[u]]=s; u=u+1], {n, 1, 1000000}]; ta
Select[Range[215000],PrimeQ[# DivisorSigma[0,#]-DivisorSigma[1,#]]&] (* Harvey P. Dale, Dec 07 2021 *)
q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; PrimeQ[n * Times @@ (e + 1) - Times @@ ((p^(e + 1) - 1)/(p - 1))]]; seq[lim_] := Module[{m1 = Floor[Sqrt[lim/2]], m2 = Floor[Sqrt[lim]/2]}, Join[{3}, Union[Select[2*Range[m1]^2, q], Select[4*Range[m2]^2, q]]]]; seq[220000] (* Amiram Eldar, Feb 14 2025 *)

PARI

isok(n) = isprime(n*numdiv(n)-sigma(n)); \\ Michel Marcus, Nov 12 2015

PARI

isok(k) = if(k % 2, k == 3, if(!issquare(k) && !issquare(2*k), 0, my(f = factor(k)); isprime(k * numdiv(f) - sigma(f)))); \\ Amiram Eldar, Feb 14 2025

Extensions

Name modified by Tom Edgar, Nov 12 2015