A123041 - OEIS

%I #21 Sep 12 2022 03:45:55

%S 1,2,3,6,14,15,30,35,42,70,78,105,190,210,357,418,570,714,910,1045,

%T 1254,1976,2090,2730,3135,4522,4674,5278,5412,5928,6270,8580,10659,

%U 12441,12628,13566,14630,15834,16770,17220,20026,21318,23374,24871,24882

%N Numbers m such that UnitarySigma(m)^2 = k*Sigma(m)*UnitaryPhi(m), for some integer k.

%C Terms which are squarefree appear on A121556.

%H Amiram Eldar, <a href="/A123041/b123041.txt">Table of n, a(n) for n = 1..4309</a> (terms below 10^10; terms 1..1000 from Donovan Johnson)

%t f[p_, e_] := (p - 1)*(p^e + 1)^2/((p^e - 1)*(p^(e + 1) - 1)); q[1] = True; q[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[25000], q] (* _Amiram Eldar_, Sep 12 2022 *)

%o (PARI) A047994(n)={ local(i,resul,rmax); if(n==1, return(1) ); i=factor(n); rmax=matsize(i)[1]; resul=1; for(r=1,rmax, resul *= i[r,1]^i[r,2]-1; ); return(resul); }

%o A034448(n)={ sumdiv(n, d, if(gcd(d, n/d)==1, d)) }

%o isA123041(n)={ local(s); s=(A034448(n))^2; if( s % (sigma(n)*A047994(n)) == 0, return(s/sigma(n)/A047994(n)), return(0) ); }

%o { for(n=1,30000, k=isA123041(n); if( k, print1(n,",") ); ); } \\ _R. J. Mathar_, Sep 27 2006

%o (PARI) is(n) = {my(f=factor(n), p=f[,1], e=f[,2]); denominator(prod(i=1, #p, (p[i] - 1)*(p[i]^e[i] + 1)^2/((p[i]^e[i] - 1)*(p[i]^(e[i] + 1) - 1)))) == 1}; \\ _Amiram Eldar_, Sep 12 2022

%Y Cf. A000203, A123042, A121556, A034448, A047994.

%K nonn

%O 1,2

%A _Yasutoshi Kohmoto_, Sep 24 2006

%E More terms from _R. J. Mathar_, Sep 27 2006