A371513 -id:A371513 - cp's OEIS frontend

A371883 a(n) is the number of divisors d of n such that d^n mod n = d.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 3, 4, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 1, 1, 3, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2

Offset: 1

Juri-Stepan Gerasimov, Apr 10 2024

nonn

1 <= a(n) < A000005(n) for n >= 2.

			a(1) = 0: 1 divides 1, but 1^1 mod 1 = 0 (not 1).
a(2) = 1: 1 divides 2, and 1^2 mod 2 = 1;
          2 divides 2, but 2^2 mod 2 = 0 (not 2).
a(6) = 2: 1 divides 6, and 1^6 mod 6 = 1;
          2 divides 6, but 2^6 mod 6 = 4 (not 2);
          3 divides 6, and 3^6 mod 6 = 3;
          6 divides 6, but 6^6 mod 6 = 0 (not 6).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005, A182816, A279024, A371513, A371884.

[#[d: d in Divisors(n) | d^n mod n eq d]: n in [1..100]];

a[n_] := DivisorSum[n, 1 &, PowerMod[#, n, n] == # &]; Array[a, 100] (* Amiram Eldar, Apr 11 2024 *)

a(n) = sumdiv(n, d, d^n % n == d); \\ Michel Marcus, Apr 20 2024

from sympy import divisors
def a(n): return sum(1 for d in divisors(n)[:-1] if pow(d, n, n) == d)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Apr 10 2024

A371884 Irregular triangle read by rows in which row n >= 2 lists the divisors d of n such that d^n mod n = d.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 4, 1, 1, 7, 1, 5, 1, 1, 1, 9, 1, 1, 5, 1, 7, 1, 11, 1, 1, 1, 1, 13, 1, 1, 4, 1, 1, 6, 10, 15, 1, 1, 1, 11, 1, 17, 1, 1, 9, 1, 1, 19, 1, 13, 1, 1, 1, 7, 21, 1, 1, 1, 9, 1, 23, 1, 1, 16, 1, 1, 25, 1, 17, 1, 13, 1, 1, 27, 1, 11, 1, 8, 1, 19, 1, 29, 1, 1, 1, 1, 31, 1, 1, 1, 5, 13

Offset: 2

Juri-Stepan Gerasimov, Apr 10 2024

			Triangle begins:
    1;
    1;
    1;
    1;
    1, 3;
    1;
    1;
    1;
    1, 5;
    1;
    1, 4;
    1;
    1, 7;
    1, 5;
    1;
    1;
    1, 9;
    1;
    1, 5;
    1, 7;
    1, 11;
    1;
    1;
    1;
    1, 13;
    1;
    1, 4;
    1;
    1, 6, 10, 15;
    ...

Robert Israel, Table of n, a(n) for n = 2..18962 (rows 2 to 10000, flattened)

Cf. A027750, A182816, A279024, A371513, A371883.

[[d: d in Divisors(n) | d^n mod n eq d]: n in [2..65]];

f:= proc(n) op(sort(convert(select(d -> d^n mod n = d, numtheory:-divisors(n)),list))) end proc:
for n from 2 to 100 do f(n) od; # Robert Israel, May 11 2025

row[n_] := Select[Divisors[n], PowerMod[#, n, n] == # &]; Array[row, 64, 2] // Flatten (* Amiram Eldar, Apr 11 2024 *)

A380393 a(n) is the least k that has exactly n proper divisors d such that (-d)^k == -d (mod k).

Original entry on oeis.org

1, 2, 6, 42, 66, 105, 2805, 561, 1365, 5005, 5565, 11305, 36465, 140505, 239785, 41041, 682465, 873145, 185185, 418285, 1683969, 2113665, 5503785, 1242241, 6697405, 8549905, 31932901, 11996985, 31260405, 30534805, 47031061, 825265, 27265161, 32306365, 55336645, 21662641, 9276085, 8964865

Offset: 0

Robert Israel, Jan 23 2025

nonn

a(n) is the least k such that A378387(k) = n.

It appears that a(n) = A371513(n) except for n = 4 and n = 6. They are certainly equal when a(n) and A371513(n) are odd, since if k is odd, (-d)^k = -d^k == -d (mod k) if and only if d^k == d (mod k).

			a(4) = 42 because 42 has 3 such divisors: (-6)^42 == 36 == -6 (mod 42), (-14)^42 == 28 == -14 (mod 42), (-21)^42 == 21 == -21 (mod 42), and no smaller number works.

Cf. A371513, A378387.

f:= proc(n) nops(select((t -> (-t)&^n + t mod n = 0), numtheory:-divisors(n) minus {n})) end proc:
N:= 30: # for a(0) .. a(N)
V:= Array(0..N): count:= 0:
for i from 1 while count < N+1 do
  v:= f(i);
  if v <= N and V[v] = 0 then V[v]:= i; count:= count+1 fi;
od:
convert(V,list);

a(n) = my(k=1); while (sumdiv(k, d, if (dMichel Marcus, Jan 24 2025

A378387(a(n)) = n.

cp's OEIS Frontend

A371883 a(n) is the number of divisors d of n such that d^n mod n = d.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A371884 Irregular triangle read by rows in which row n >= 2 lists the divisors d of n such that d^n mod n = d.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

A380393 a(n) is the least k that has exactly n proper divisors d such that (-d)^k == -d (mod k).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Formula