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

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

Author

Vladislav Shubin, Sep 20 2020

Keywords

• nonn

References

• 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.

Links

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

Crossrefs

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

Programs

Mathematica

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]

Formula

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

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