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A123041
Numbers m such that UnitarySigma(m)^2 = k*Sigma(m)*UnitaryPhi(m), for some integer k.
2
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
KEYWORD
nonn
AUTHOR
Yasutoshi Kohmoto, Sep 24 2006
EXTENSIONS
More terms from R. J. Mathar, Sep 27 2006
STATUS
approved