A121556 -id:A121556 - cp's OEIS frontend

A121647 Sequence gives values of sigma(n)^2 divided by UnitarySigma(n)*UnitaryPhi(n) associated with n in A121556.

1, 3, 2, 6, 4, 3, 9, 2, 8, 6, 7, 4, 5, 12, 7, 3, 4, 10, 9, 7, 2, 5, 8, 6, 5, 14, 4, 10, 7, 5, 7, 5, 12, 10, 4, 3, 9, 14, 3, 10, 8, 10, 11, 15, 4, 9, 4, 2, 9, 3, 5, 6, 4, 7, 2, 8, 5, 15, 4, 11, 4, 5, 10, 16, 7, 10, 6, 13, 9, 2, 6, 7, 5, 4, 4, 6, 2, 8, 2, 15, 5, 8, 4, 11, 10, 12, 8

Offset: 1

Yasutoshi Kohmoto Sep 12 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000203, A034448, A047994, A121556.

f[p_, e_] := (p^(e+1)-1)^2/(p-1)^2/(p^(2*e)-1); ratio[1] = 1; ratio[n_] := Times @@ (f @@@ FactorInteger[n]); Select[ratio /@ Range[10^5], IntegerQ] (* Amiram Eldar, Dec 11 2019 *)

a(n) = f(A121556(n)), where f(m) = A000203(m)^2/(A034448(m)*A047994(m)). - Amiram Eldar, Dec 11 2019

Extended with the aid of b121556.txt. - R. J. Mathar, Sep 04 2018

A122839 Numbers such that Sigma(m)UnitarySigma(m)= kUnitaryPhi(m)^2, for some integer k.

Original entry on oeis.org

1, 2, 3, 6, 14, 15, 30, 35, 42, 60, 70, 78, 105, 190, 210, 312, 357, 418, 570, 714, 910, 1045, 1254, 1428, 2090, 2730, 3135, 3640, 4522, 4674, 5278, 6270, 10659, 10920, 12441, 13566, 14630, 15834, 16770, 18696, 20026, 21318, 23374, 24871, 24882, 24969, 25070, 25714, 27170

Offset: 1

Yasutoshi Kohmoto, Oct 23 2006

nonn

Terms which are squarefree appear on A121556.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000203, A034448, A047994, A121556.

isA122839 := proc(m)
    A047994(m) ;
    modp(numtheory[sigma](m)*A034448(m),%^2) = 0 ;
end proc:
for m from 1 do
    if isA122839(m) then
        printf("%a,\n", m) ;
    end if;
end do: # R. J. Mathar, Sep 04 2018

f[p_, e_] := (p^(e+1)-1)*(p^e+1)/(p-1)/(p^e-1)^2; seqQ[1] = True; seqQ[n_] := IntegerQ [Times @@ (f @@@ FactorInteger[n])]; Select[Range[27170], seqQ] (* Amiram Eldar, Dec 11 2019 *)

isok(m) = ((sigma(m)*usigma(m)) % uphi(m)^2) == 0; \\ Michel Marcus, Jan 24 2019

More terms from Michel Marcus, Jan 24 2019

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

Original entry on oeis.org

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

Yasutoshi Kohmoto, Sep 24 2006

nonn

Terms which are squarefree appear on A121556.

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

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

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 *)

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

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

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

cp's OEIS Frontend

A121647 Sequence gives values of sigma(n)^2 divided by UnitarySigma(n)*UnitaryPhi(n) associated with n in A121556.

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A122839 Numbers such that Sigma(m)UnitarySigma(m)= kUnitaryPhi(m)^2, for some integer k.

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

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Author

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Mathematica

PARI

PARI

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cp's OEIS Frontend

A121647 Sequence gives values of sigma(n)^2 divided by UnitarySigma(n)*UnitaryPhi(n) associated with n in A121556.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A122839 Numbers such that Sigma(m)*UnitarySigma(m)= k*UnitaryPhi(m)^2, for some integer k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A122839 Numbers such that Sigma(m)UnitarySigma(m)= kUnitaryPhi(m)^2, for some integer k.

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