A394407 - OEIS

%I #10 Apr 01 2026 16:03:34

%S 1,2,4,5,7,11,13,14,16,17,19,20,22,23,25,27,29,31,32,34,37,41,43,44,

%T 46,47,53,54,55,58,59,61,62,64,65,67,68,71,73,79,80,81,83,85,86,89,92,

%U 94,97,100,101,103,106,107,109,112,113,115,116,118,119,121,125,127

%N Positive integers k such that there is no m < k with phi(m)/tau(m) = phi(k)/tau(k).

%C Conjecture: All primes p != 3 are in the sequence; equivalently, for every prime p != 3 there is no m < p with phi(m)/tau(m) = phi(p)/tau(p) = (p - 1)/2; equivalently, the sequence is a supersequence of A045344. Verified for all primes p < 10^6.

%H Felix Huber, <a href="/A394407/b394407.txt">Table of n, a(n) for n = 1..10000</a>

%e 14 is a term, since there is no m < 14 with phi(m)/tau(m) = phi(14)/tau(14) = 6/4 = 3/2.

%e 3 is not a term, since phi(1)/tau(1) = 1/1 = 2/2 = phi(3)/tau(3) and 1 < 3.

%p with(NumberTheory):

%p A394407List := proc(N) # All terms <= N

%p     local a, k, l, r, t;

%p     a := [];

%p     t := table();

%p     for k to N do

%p         r := phi(k)/tau(k);

%p         l := [numer(r), denom(r)];

%p         if not assigned(t[l]) then

%p             t[l] := true;

%p             a := [op(a), k]

%p         end if

%p     end do;

%p     a

%p end proc:

%p A394407List(127);

%t fpt[x_]:=EulerPhi[x]/DivisorSigma[0,x];q[k_]:=AllTrue[fpt/@Range[k-1],#!=fpt[k]&];Select[Range[127],q] (* _James C. McMahon_, Apr 01 2026 *)

%Y Cf. A000005, A000010, A045344, A094872, A175667, A279287, A279288.

%K nonn,easy

%O 1,2

%A _Felix Huber_, Mar 28 2026