Vladislav Shubin - cp's OEIS frontend

User: Vladislav Shubin

Vladislav Shubin has authored 4 sequences.

A355059 a(n) is the positive integer such that psi(A355045(n)) = rad(A355045(n))^a(n), where psi(k) = A001615(k) and rad(k) = A007947(k).

2, 2, 3, 5, 4, 6, 3, 5, 5, 4, 7, 6, 6, 6, 6, 4, 5, 8, 5, 5, 7, 7, 7, 4, 6, 4, 9, 4, 6, 6, 9, 6, 6, 8, 4, 8, 8, 8, 8, 5, 5, 7, 8, 7, 5, 6, 5, 10, 7, 7, 7, 7, 5, 7, 7, 7, 5, 7, 9, 7, 7, 6, 9, 9, 9, 5, 9, 7, 6, 7, 6, 6, 9, 6, 7, 9, 8, 7, 6, 8, 8, 6, 6, 11, 8, 8

Offset: 1

Vladislav Shubin, Jun 16 2022

nonn

Vladislav Shubin, Table of n, a(n) for n = 1..1000

Cf. A001615, A007947, A355045.

DedekindPsi[n_] := n * Product[(1 + 1/i), {i, FactorInteger[n][[All, 1]]}];
bound = 86
For[s = 1, s <= bound, s++,
   If[s == 1, Print["m   =   ", 1]; s = s + 1;];
   Q = Prime[s];
   InitialArray = FactorInteger[If[Q != 3, 3*(Q + 1), 2]];
   For[i = 1, i <= Length[InitialArray] - 1, i++,
        CurrentArray = FactorInteger[InitialArray[[-i, 1]] + 1] ~Join~ InitialArray;
        InitialArray = FactorInteger[Product[CurrentArray[[k, 1]] ^ CurrentArray[[k, 2]], {k, 1, Length[CurrentArray]}]];
   ];
        InitialArray = InitialArray~Join~{{Q, 0}};
        m = Max[InitialArray[[All, 2]]];
   If[Q == 3, m = m + 1];
   Print["m   =   ", m]
];

A355045 a(n) is the least positive integer k which is a multiple of prime(n) such that for some m >= 0, psi(k) = rad(k)^m, where psi(k) = A001615(k) and rad(k) = A007947(k).

Original entry on oeis.org

18, 18, 11250, 57177414, 8696754, 10763393803185114, 501126, 23816977256250, 23981814018, 230750426250, 3730545397766934, 33914855378546706844968750, 11135545745963323734, 234030019748505421122, 246218836545018, 5018345916

Offset: 1

Vladislav Shubin, Jun 16 2022

nonn

Vladislav Shubin, Table of n, a(n) for n = 1..1000

Cf. A001615, A007947.

DedekindPsi[n_] :=
  n * Product[(1 + 1/i), {i, FactorInteger[n][[All, 1]]}];
For[s = 2, s <= 49, s++,
   If[s == 1, Print["n   =   ", 18]; s = s + 1;];
   Q = 1*Prime[s];
InitialArray = FactorInteger[If[Q != 3, 3*(Q + 1), 2]];
For[i = 1, i <= Length[InitialArray] - 1, i++,
  CurrentArray =
   FactorInteger[InitialArray[[-i, 1]] + 1] ~Join~ InitialArray;
      InitialArray =
   FactorInteger[Product[CurrentArray[[k, 1]] ^ CurrentArray[[k, 2]], {k, 1,
      Length[CurrentArray]}]];
  ];
InitialArray = InitialArray ~Join~ {{Q, 0}};
m = Max[InitialArray[[All, 2]]];
n = Product[Power[InitialArray[[k, 1]], m - InitialArray[[k, 2]] + 1], {k, 1,
    Length[InitialArray]}];
If[Q ==  3, m = m + 1];
Print["n   =   " n]];

A337776 a(n) is the order of A337775(n) (as defined in that sequence).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 2, 4, 4, 4, 3, 5, 3, 5, 5, 5, 5, 5, 5, 4, 4, 6, 6, 6, 4, 6, 6, 3, 6, 3, 6, 7, 6, 5, 4, 5, 4, 7, 5, 7, 5, 5, 7, 4, 5, 5, 4, 7, 4, 7, 7, 7, 3, 8, 7, 7, 4, 7, 7, 7, 6, 6, 5, 6, 6, 5, 6, 6, 6, 8, 8, 6, 6, 4, 6, 8, 6, 6, 8, 6, 6, 6, 5, 5, 8, 8, 5, 6, 6, 8, 8, 5, 5, 8, 4, 5

Offset: 1

Vladislav Shubin, Sep 20 2020

nonn

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 108, p. 38, Ellipses, Paris 2008.
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problème 745 ; pp 95; 317-8, Ellipses Paris 2004.

J.-M. De Koninck, When the Totient Is the Product of the Squared Prime Divisors: Problem 10966, Amer. Math. Monthly, 111 (2004), p. 536.

Cf. A000010 (phi), A000040 (prime), A007947 (rad), A337775.

nn = 97;
Sar = Table[0, {nn}]; Sar[[1]] = 2;
(*It is a list oh the sequence A337775*)
OrdSar = Table[0, {nn}]; OrdSar[[1]] = 0;
(*It is a sequence A337776 - the orders of members in sequence A337775*) For[Index = 2, Index <= nn, Index++,
  InitialPrime = Prime[Index];
  InitialInteger = InitialPrime - 1;
  InitialArray = FactorInteger[InitialInteger];
  For[i = 1, i <= Length[InitialArray], i++,
   CurrentArray =
    FactorInteger[InitialArray[[-i, 1]] - 1] ~Join~ InitialArray;
   InitialInterger =
    Product[CurrentArray[[k, 1]] ^ CurrentArray[[k, 2]], {k, 1,
      Length[CurrentArray]}];
     InitialArray = FactorInteger[InitialInterger];
   ];
  InitialArray = InitialArray ~Join~ {{InitialPrime, 0}};
  Ord = Max[InitialArray[[All, 2]]];
  Lint = Product[
    Power[InitialArray[[k, 1]], Ord - InitialArray[[k, 2]] + 1], {k,
     1, Length[InitialArray]}];
  radn = Product[InitialArray[[k, 1]], {k, 1, Length[InitialArray]}];
  Sar[[Index]] = Lint;
  OrdSar[[Index]] = Ord;
  ];
Print["Sar=  ", Sar]
Print["OrdSar=  ", OrdSar]

rad(A337775(n))^a(n) = phi(A337775(n)).

a(n) = log(phi(A337775(n))) / log(rad(A337775(n))). - Andrew Howroyd, Sep 21 2020

A337775 a(n) is the least natural k which is a multiple of prime(n) such that for some m >= 0, phi(k) = rad(k)^m, where phi(k) = A000010(k) and rad(k) = A007947(k).

Original entry on oeis.org

2, 18, 250, 6174, 3660250, 1542294, 2839714, 41154, 117793122328750, 7978057537338, 2898701538750, 33734898, 29688151506250, 21107677374, 69834458642125879757481250, 3999523458421521342

Offset: 1

Vladislav Shubin, Sep 20 2020

nonn

The number m mentioned above is usually referred to as the order of the corresponding number a(n). The sequence of these orders is in A337776.
The algorithm suggested here for the calculation of a(n) starts its work from prime(n).
Numbers k such that phi(k) = rad(k)^m with m >= 1 are given in A211413. - Andrew Howroyd, Sep 21 2020

			For n=12 the initial prime is prime(12) = 37 and a(12) = 33734898 because phi(33734898) = 10941048, rad(33734898) = 222 and 222^3 = 10941048 and there is no smaller number satisfying the requirements. The order of a(12) is 3.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 108, p. 38, Ellipses, Paris 2008.
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problème 745 ; pp 95; 317-8, Ellipses Paris 2004.

J.-M. De Koninck, When the Totient Is the Product of the Squared Prime Divisors: Problem 10966, Amer. Math. Monthly, 111 (2004), p. 536.

Cf. A000010 (phi), A000040 (prime), A007947 (rad), A023503, A024619, A105261, A211413.

nn = 16;
Sar = Table[0, {nn}]; Sar[[1]] = 2;
(*It is a list oh the sequence A337775*)
OrdSar = Table[0, {nn}]; OrdSar[[1]] = 0;
(*It is a sequence A337776 - the orders of members in sequence A337775*) For[Index = 2, Index <= nn, Index++,
  InitialPrime = Prime[Index];
  InitialInteger = InitialPrime - 1;
  InitialArray = FactorInteger[InitialInteger];
  For[i = 1, i <= Length[InitialArray], i++,
   CurrentArray =
    FactorInteger[InitialArray[[-i, 1]] - 1] ~Join~ InitialArray;
   InitialInterger =
    Product[CurrentArray[[k, 1]] ^ CurrentArray[[k, 2]], {k, 1,
      Length[CurrentArray]}];
     InitialArray = FactorInteger[InitialInterger];
   ];
  InitialArray = InitialArray ~Join~ {{InitialPrime, 0}};
  Ord = Max[InitialArray[[All, 2]]];
  Lint = Product[
    Power[InitialArray[[k, 1]], Ord - InitialArray[[k, 2]] + 1], {k,
     1, Length[InitialArray]}];
  radn = Product[InitialArray[[k, 1]], {k, 1, Length[InitialArray]}];
  Sar[[Index]] = Lint;
  OrdSar[[Index]] = Ord;
  ];
Print["Sar=  ", Sar]
Print["OrdSar=  ", OrdSar]

rad(n) = factorback(factorint(n)[, 1]);
isok(k) = my(phik=eulerphi(k), radk=rad(k), x=logint(phik, radk)); radk^x == phik;
a(n) = {my(p=prime(n), k=p); while (!isok(k), k+=p); k;} \\ Michel Marcus, Sep 23 2020

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User: Vladislav Shubin

A355059 a(n) is the positive integer such that psi(A355045(n)) = rad(A355045(n))^a(n), where psi(k) = A001615(k) and rad(k) = A007947(k).

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A355045 a(n) is the least positive integer k which is a multiple of prime(n) such that for some m >= 0, psi(k) = rad(k)^m, where psi(k) = A001615(k) and rad(k) = A007947(k).

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A337776 a(n) is the order of A337775(n) (as defined in that sequence).

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A337775 a(n) is the least natural k which is a multiple of prime(n) such that for some m >= 0, phi(k) = rad(k)^m, where phi(k) = A000010(k) and rad(k) = A007947(k).

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