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

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

1,1

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

Formula

A187680(a(n)) = 1.

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 *)

Prog

(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

Crossrefs

Cf. A000203, A007955, A187680.

Sequence in context: A206920 A108612 A065741 * A279456 A069560 A075494

Adjacent sequences: A188058 A188059 A188060 * A188062 A188063 A188064

Keyword

nonn

Author

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

Extensions

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

Status

approved