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
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
KEYWORD
nonn
AUTHOR
Vladislav Shubin, Sep 20 2020
STATUS
approved