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A372772 a(n) is the number of divisors d of n such that d^n mod n = k, where k is also a divisor of n.

%I A372772 #22 May 27 2024 23:10:41
%S A372772 0,1,1,1,1,2,1,1,1,2,1,3,1,2,2,1,1,3,1,2,2,2,1,1,1,2,1,2,1,4,1,1,2,2,
%T A372772 1,3,1,2,2,1,1,3,1,1,2,2,1,5,1,3,2,2,1,4,2,2,2,2,1,1,1,2,1,1,3,4,1,2,
%U A372772 2,2,1,3,1,2,2,1,1,3,1,5,1,2,1,4,2,2,2,1,1,5,2,1,2,2,2,1,1,3,1,3
%N A372772 a(n) is the number of divisors d of n such that d^n mod n = k, where k is also a divisor of n.
%H A372772 Antti Karttunen, <a href="/A372772/b372772.txt">Table of n, a(n) for n = 1..16384</a>
%e A372772 a(12) = 3: 1 divides 12, and 1^12 mod 12 = 1;
%e A372772            2 divides 12, and 2^12 mod 12 = 4;
%e A372772            3 divides 12, but 3^12 mod 12 = 9 (not a divisor of 12);
%e A372772            4 divides 12, and 4^12 mod 12 = 4;
%e A372772            6 divides 12, but 6^12 mod 12 = 0 (not a divisor of 12);
%e A372772            12 divides 12, but 12^12 mod 12 = 0 (not a divisor of 12).
%t A372772 a[n_] := DivisorSum[n, 1 &, (m = PowerMod[#, n, n]) > 0 && Divisible[n, m] &]; Array[a, 100] (* _Amiram Eldar_, May 13 2024 *)
%o A372772 (Magma) [&+[#[d: d in Divisors(n) | d^n mod n eq k and n mod k eq 0]: k in [1..n]]: n in [1..100]];
%o A372772 (PARI) A372772(n) = { my(k); sumdiv(n, d, k=lift(Mod(d^n,n)); k > 0 && 0==(n%k)); }; \\ _Antti Karttunen_, May 13 2024
%o A372772 (Python)
%o A372772 from sympy import divisors
%o A372772 def a(n):
%o A372772     divs = set(divisors(n)[:-1])
%o A372772     return sum(1 for d in divs if pow(d, n, n) in divs)
%o A372772 print([a(n) for n in range(1, 101)]) # _Michael S. Branicky_, May 13 2024
%Y A372772 Cf. A371883.
%K A372772 nonn
%O A372772 1,6
%A A372772 _Juri-Stepan Gerasimov_, May 12 2024