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).

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

Offset

1,1

Comments

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

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

Table of n, a(n) for n=1..16.

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

Example

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.

Mathematica

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]

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A265452 A121429 A368466 * A276364 A109517 A213643

Adjacent sequences: A337772 A337773 A337774 * A337776 A337777 A337778

Keyword

nonn

Author

Vladislav Shubin, Sep 20 2020

Status

approved