A123041 Numbers m such that UnitarySigma(m)^2 = k*Sigma(m)*UnitaryPhi(m), for some integer k.

1, 2, 3, 6, 14, 15, 30, 35, 42, 70, 78, 105, 190, 210, 357, 418, 570, 714, 910, 1045, 1254, 1976, 2090, 2730, 3135, 4522, 4674, 5278, 5412, 5928, 6270, 8580, 10659, 12441, 12628, 13566, 14630, 15834, 16770, 17220, 20026, 21318, 23374, 24871, 24882

Offset

1,2

Comments

Terms which are squarefree appear on A121556.

Links

Amiram Eldar, Table of n, a(n) for n = 1..4309 (terms below 10^10; terms 1..1000 from Donovan Johnson)

Mathematica

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 *)

Prog

(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); }
A034448(n)={ sumdiv(n, d, if(gcd(d, n/d)==1, d)) }
isA123041(n)={ local(s); s=(A034448(n))^2; if( s % (sigma(n)*A047994(n)) == 0, return(s/sigma(n)/A047994(n)), return(0) ); }
{ for(n=1, 30000, k=isA123041(n); if( k, print1(n, ", ") ); ); } \\ R. J. Mathar, Sep 27 2006
(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

Crossrefs

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

Sequence in context: A353869 A258077 A121556 * A078557 A005537 A306600

Adjacent sequences: A123038 A123039 A123040 * A123042 A123043 A123044

Keyword

nonn

Author

Yasutoshi Kohmoto, Sep 24 2006

Extensions

More terms from R. J. Mathar, Sep 27 2006

Status

approved