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
Keywords
Examples
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.
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.
Programs
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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] -
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
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