A188061 Numbers k such that (product of divisors of k) == 1 (mod sum of divisors of k).

4, 9, 16, 25, 49, 55, 64, 81, 121, 161, 169, 209, 256, 289, 351, 361, 529, 551, 625, 649, 729, 841, 961, 1024, 1079, 1189, 1369, 1407, 1443, 1681, 1849, 2015, 2209, 2289, 2401, 2809, 2849, 2915, 2975, 3401, 3481, 3721, 3857, 4096, 4489, 4599, 4887, 5041, 5329, 6049, 6241, 6319, 6561, 6889, 6993, 7921, 8569, 9409, 9701

Offset: 1

Author

Juri-Stepan Gerasimov, Mar 20 2011, Jun 18 2011

Keywords

• nonn

Comments

This sequence includes every number of the form p^(2n), where p is a prime. Other semiprime members include 55, 161, 209, 551, 649, 1079, 1189, 3401, 6049, 6319, 9701. Are there infinitely many nonsquare semiprimes in the sequence? Is there some simpler property of primes p and q that puts pq in this sequence?

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000203, A007955, A187680.

Programs

Mathematica

mptQ[n_]:=Module[{dn=Divisors[n]},Mod[Times@@dn,Total[dn]]==1]; Join[{1},Select[Range[10000],mptQ]]  (* Harvey P. Dale, Mar 28 2011 *)

PARI

proddiv(n)=local(t);t=numdiv(n);if(t%2==0,n^(t\2),sqrtint(n)^t)
for(n=1,10000,if(Mod(proddiv(n),sigma(n))==1,print1(n",")))

Python

from gmpy2 import powmod, is_square, isqrt
from sympy import divisor_sigma
A188061_list = [n for n in range(1,10**4) if powmod(isqrt(n) if is_square(n) else n, int(divisor_sigma(n,0))//(1 if is_square(n) else 2), int(divisor_sigma(n,1))) == 1] # Chai Wah Wu, Mar 10 2016

Formula

A187680(a(n)) = 1.

Extensions

More terms from Franklin T. Adams-Watters, Mar 21 2011