A394407 Positive integers k such that there is no m < k with phi(m)/tau(m) = phi(k)/tau(k).

1, 2, 4, 5, 7, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 27, 29, 31, 32, 34, 37, 41, 43, 44, 46, 47, 53, 54, 55, 58, 59, 61, 62, 64, 65, 67, 68, 71, 73, 79, 80, 81, 83, 85, 86, 89, 92, 94, 97, 100, 101, 103, 106, 107, 109, 112, 113, 115, 116, 118, 119, 121, 125, 127

Offset

1,2

Comments

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.

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Example

14 is a term, since there is no m < 14 with phi(m)/tau(m) = phi(14)/tau(14) = 6/4 = 3/2.
3 is not a term, since phi(1)/tau(1) = 1/1 = 2/2 = phi(3)/tau(3) and 1 < 3.

Maple

with(NumberTheory):
A394407List := proc(N) # All terms <= N
    local a, k, l, r, t;
    a := [];
    t := table();
    for k to N do
        r := phi(k)/tau(k);
        l := [numer(r), denom(r)];
        if not assigned(t[l]) then
            t[l] := true;
            a := [op(a), k]
        end if
    end do;
    a
end proc:
A394407List(127);

Mathematica

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 *)

Crossrefs

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

Sequence in context: A389609 A255850 A108464 * A128815 A056527 A147991

Adjacent sequences: A394404 A394405 A394406 * A394408 A394409 A394410

Keyword

nonn,easy

Author

Felix Huber, Mar 28 2026

Status

approved