A291932 - OEIS

%I #40 Sep 21 2017 01:31:29

%S 2,3,95,5,143,7,319,323,559,11,117317,13,1007,899,1919,17,201983,19,

%T 441283,1763,394697,23,4031,5249,2911,3239,23519,29,3599,31,1796647,

%U 979801,8159,5459,5183,37,1550047,10763,8639,41,2709037,43,10207,9179,101567,47,12218993,9701,13199,10403,4018073,53

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

%C 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

%C a(102) > 100000000. - _Robert G. Wilson v_, Sep 08 2017

%C a(102) = 8759437837. - _Giovanni Resta_, Sep 11 2017

%C a(108) > 2550000000. - _Robert G. Wilson v_, Sep 20 2017

%H Robert G. Wilson v, <a href="/A291932/b291932.txt">Table of n, a(n) for n = 2..107</a>

%F a(p) = p for all primes p.

%e 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.

%p N:= 10^7: # to get all terms before the first with a(n) > N

%p M:= nextprime(N):

%p A:= Vector(M):

%p R:= proc(n) mul((i[1]+1)/(i[1]-1),i=ifactors(n)[2]) end proc:

%p for k from 2 to N do

%p r:= R(k);

%p n:= (r+1)/(r-1);

%p if n::integer and n <= M and A[n] = 0 then

%p   A[n]:= k;

%p fi

%p od:

%p m:=min(select(t -> A[t]=0, [$2..M]))-1:

%p seq(A[i],i=2..m); # _Robert Israel_, Sep 06 2017

%t 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 *)

%o (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

%Y Cf. A000010, A001615.

%K nonn

%O 2,1

%A _Altug Alkan_, Sep 06 2017