A064498 - OEIS

A064498

Numbers k such that the sum of unitary divisors of k^2 is a square.

%I #30 Apr 13 2024 15:24:55

%S 1,42,120,156,246,287,1434,1673,2016,5256,9799,11808,18330,19740,

%T 21385,34440,39990,44772,45990,46655,57270,60156,66815,68832,102648,

%U 115620,125255,149472,156570,170820,182665,195510,200760,208182,223944,224196

%N Numbers k such that the sum of unitary divisors of k^2 is a square.

%H Amiram Eldar, <a href="/A064498/b064498.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..100 from Harry J. Smith)

%t sudsQ[n_]:=Module[{uds=Sort[Flatten[Outer[Times,Sequence@@({1,#}&/@ Power@@@FactorInteger[n^2])]]]},IntegerQ[Sqrt[Total[uds]]]]; Join[{1}, Select[Range[230000],sudsQ]] (* _Harvey P. Dale_, Dec 09 2011 *)

%o (PARI) {usigma(n, s=1, fac, i)= fac=factor(n); for(i=1,matsize(fac)[1], s=s*(1+fac[i,1]^fac[i,2]); ); return(s); }

%o for(n=1,10^6, if(issquare(usigma(n^2)),print1(n," ")))

%o (PARI) usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }

%o { n=0; for (m=1, 10^9, if (issquare(usigma(m^2)), write("b064498.txt", n++, " ", m); if (n==100, break)) ) } \\ _Harry J. Smith_, Sep 16 2009

%Y Cf. A034448 (usigma), A008847 (similar, with sigma).

%K nonn

%O 1,2

%A _Jason Earls_, Oct 05 2001