A272442 Numbers n such that the average of the positive divisors of n^2 is a square.

1, 313, 4077, 42217, 75848, 1276101, 2468936, 8105257, 13213921, 23740424, 76136648, 172118709, 249948601, 296474857, 309232296, 772776968, 1236329761, 1254551521, 2288805793, 2372141576, 2410951561, 2536945441, 3202075016, 10065852072

Offset: 1

Author

Altug Alkan, Apr 29 2016

Keywords

• nonn

Comments

The values of n are 1, 313, 3^3*151, 7*37*163, 2^3*19*499, 3^3*151*313, 2^3*19*37*439, 37*439*499, ...

Corresponding n^2 values are 1, 97969, 16621929, 1782275089, ...

The next two primes in the sequence (after 313 and 2288805793) are 86136917171761 and 23666840016182721134955337. - Giovanni Resta, Apr 26 2017

Examples

			313 is a term because the divisors of 313^2 are 1, 313 and 313^2. Average of 1, 313 and 313^2 is 32761 that is 181^2.
		

Crossrefs

Cf. A000290, A003601, A144695.

Programs

Mathematica

Select[Range[10^5], IntegerQ@ Sqrt@ Mean@ Divisors[#^2] &] (* Michael De Vlieger, Apr 29 2016 *)

PARI

lista(nn) = for(n=1, nn, if(sigma(n^2) % numdiv(n^2) == 0 && issquare(sigma(n^2)/numdiv(n^2)), print1(n, ", ")));

PARI

is(n)=my(f=factor(n),s,d); f[,2]*=2; s=sigma(f); d=numdiv(f); s%d==0 && issquare(s/d) \\ Charles R Greathouse IV, Apr 29 2016

Extensions

a(10)-a(16) from Charles R Greathouse IV, Apr 29 2016

a(17)-a(24) from Giovanni Resta, Apr 26 2017