A340273 a(n) is the number of divisors d of n such that phi(n)/phi(lpf(n)) mod phi(n)/phi(d) = 0, where phi is Euler's totient function (A000010), and lpf(n) is the least prime factor of n (A020639).

1, 2, 1, 3, 1, 4, 1, 4, 2, 4, 1, 6, 1, 4, 3, 5, 1, 6, 1, 6, 3, 4, 1, 8, 2, 4, 3, 6, 1, 8, 1, 6, 3, 4, 2, 9, 1, 4, 3, 8, 1, 8, 1, 6, 5, 4, 1, 10, 2, 6, 3, 6, 1, 8, 2, 8, 3, 4, 1, 12, 1, 4, 5, 7, 3, 8, 1, 6, 3, 8, 1, 12, 1, 4, 5, 6, 2, 8, 1, 10, 4, 4, 1, 12, 3, 4

Offset: 1

Author

Maxim Karimov, Jan 02 2021

Keywords

• nonn

Comments

This equivalence criterion splits the divisor set of n into two types of divisors and can be used to compute the number of links of length k on the set of Fibonacci necklaces (A000358) of length n. This counting is a combinatorial problem over the positive integers.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000005, A000027, A000358.

Programs

MATLAB

n=100;
A=[];
for i=1:n
    d=divisors(i);
    t=0;
    for j=1:size(d,2)
        if checkCD(i,d(j))==1
            t=t+1;
        end
    end
    A=[A t];
end
function [res] = checkCD(n,d)
    if mod(n,d)==0 && mod(totient(n)/totient(min(factor(n))),totient(n)/totient(d))==0
        res=1;
    else
        res=0;
    end
end
function [res] = totient(n)
res=0;
    for i=1:n
        if gcd(i,n)==1
            res=res+1;
        end
    end
end

Maple

with(numtheory):
a:= n-> `if`(n=1, 1, (f-> nops(select(d-> irem(phi(n)/phi(f),
         phi(n)/phi(d))=0, divisors(n))))(min(factorset(n)))):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 12 2021

Mathematica

Table[Function[{e, f}, DivisorSum[n, 1 &, Mod[e, f/EulerPhi[#]] == 0 &]] @@ {#2/#1, #2} & @@ {EulerPhi[FactorInteger[n][[1, 1]]], EulerPhi[n]}, {n, 86}] (* Michael De Vlieger, Feb 12 2021 *)

PARI

lpf(n) = if (n==1, 1, factor(n)[1,1]);
a(n) = my(lp = lpf(n), t = eulerphi(n)); sumdiv(n, d, Mod(t/eulerphi(lp), t/eulerphi(d)) == 0); \\ Michel Marcus, Jan 03 2021