A291932 a(n) is the smallest k such that (n+1)*phi(k) = (n-1)*psi(k).

2, 3, 95, 5, 143, 7, 319, 323, 559, 11, 117317, 13, 1007, 899, 1919, 17, 201983, 19, 441283, 1763, 394697, 23, 4031, 5249, 2911, 3239, 23519, 29, 3599, 31, 1796647, 979801, 8159, 5459, 5183, 37, 1550047, 10763, 8639, 41, 2709037, 43, 10207, 9179, 101567, 47, 12218993, 9701, 13199, 10403, 4018073, 53

Offset: 2

Author

Altug Alkan, Sep 06 2017

Keywords

• nonn

Comments

Least k such that Product_{p|k} (p+1)/(p-1) = (n+1)/(n-1). As a result, all terms are squarefree. - Charles R Greathouse IV, Sep 06 2017

a(102) > 100000000. - Robert G. Wilson v, Sep 08 2017

a(102) = 8759437837. - Giovanni Resta, Sep 11 2017

a(108) > 2550000000. - Robert G. Wilson v, Sep 20 2017

Examples

			a(4) = 95 = 5*19 because (psi(5*19) + phi(5*19)) / (psi(5*19) - phi(5*19)) = (6*20 + 4*18) / (6*20 - 4*18) = 4 and 95 is the least number with this property.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 2..107

Crossrefs

Cf. A000010, A001615.

Programs

Maple

N:= 10^7: # to get all terms before the first with a(n) > N
M:= nextprime(N):
A:= Vector(M):
R:= proc(n) mul((i[1]+1)/(i[1]-1),i=ifactors(n)[2]) end proc:
for k from 2 to N do
r:= R(k);
n:= (r+1)/(r-1);
if n::integer and n <= M and A[n] = 0 then
  A[n]:= k;
fi
od:
m:=min(select(t -> A[t]=0, [$2..M]))-1:
seq(A[i],i=2..m); # Robert Israel, Sep 06 2017

Mathematica

psi[n_] := If[n < 1, 0, n Sum[ MoebiusMu[d]^2/d, {d, Divisors@ n}]]; f[n_] := Block[{k = 1}, While[(n + 1)*EulerPhi[k] != (n - 1)*psi[k], k++]; k]; Array[f, 52, 2] (* Robert G. Wilson v, Sep 06 2017 *)

PARI

a(n)=my(target=2/(n-1)+1,start=n,end=10*n,f); while(1, forfactored(k=start,end, f=k[2][,1]; if(vecmax(k[2][,2])==1 && prod(i=1,#f, 2/(f[i]-1)+1)==target, return(k[1]))); start=end+1; end*=2) \\ Charles R Greathouse IV, Sep 06 2017

Formula

a(p) = p for all primes p.