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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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 *)

PROG

(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

(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

CROSSREFS

Cf. A000005, A000027, A000358.

Sequence in context: A094741 A285577 A324392 * A029234 A102613 A097019

Adjacent sequences:  A340270 A340271 A340272 * A340274 A340275 A340276

KEYWORD

nonn

AUTHOR

Maxim Karimov, Jan 02 2021

STATUS

approved

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Last modified January 24 09:02 EST 2022. Contains 350534 sequences. (Running on oeis4.)