A390872 a(n) is the number of positive integers k such that tau(k) + phi(k) = n.

0, 1, 1, 1, 1, 2, 0, 3, 1, 2, 0, 3, 1, 3, 0, 4, 0, 2, 0, 3, 1, 3, 1, 3, 0, 5, 0, 4, 0, 3, 0, 5, 0, 1, 0, 4, 0, 2, 1, 3, 0, 4, 0, 5, 1, 2, 0, 8, 1, 2, 0, 4, 0, 1, 0, 4, 0, 1, 1, 4, 0, 3, 1, 7, 0, 3, 0, 4, 0, 1, 0, 5, 0, 2, 0, 6, 0, 3, 0, 6, 0, 1, 0, 5, 0, 2, 0, 5

Offset

1,6

Comments

a(n) is the number of positive integers k such that A000005(k) + A000010(k) = n.

A safe upper bound for moderate to large k is ceiling(n*log(n)^2), since phi(k) >= k/log(log(k)) and tau(k) >= 2.

Links

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

Eric Weisstein's World of Mathematics, Totient Function

Example

a(8) = 3 because tau(7) + phi(7) = tau(8) + phi(8) = tau(10) + phi(10) = 8.

Maple

with(NumberTheory):
A390872 := proc(N)
    local k, V, v;
    V := Vector(1..N);
    for k to max(20, ceil(N*log(N)^2)) do
        v := tau(k) + phi(k);
        if v <= N then
            V[v] := V[v] + 1
        fi
    od;
    op(convert(V, list));
end proc:
A390872(88);

Mathematica

a[n_]:=Length[Select[Range[Ceiling[n*Log[n]^2]], DivisorSigma[0, #]+EulerPhi[#]==n&]]; Array[a, 88] (* James C. McMahon, Nov 22 2025 *)

Crossrefs

Cf. A000005, A000010, A054973, A100966.

Sequence in context: A382993 A071465 A333409 * A197117 A343879 A275387

Adjacent sequences: A390869 A390870 A390871 * A390873 A390874 A390875

Keyword

nonn

Author

Felix Huber, Nov 22 2025

Status

approved