A371883 -id:A371883 - cp's OEIS frontend

A371513 a(n) is the smallest number m with n divisors d such that d^m mod m = d.

1, 2, 6, 42, 30, 105, 910, 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

Offset: 0

Juri-Stepan Gerasimov, Apr 10 2024

nonn

			a(0) = 1 with divisors {};
a(1) = 2 with divisor {1};
a(2) = 6 with divisors {1, 3};
a(3) = 42 with divisors {1, 7, 21};
a(4) = 30 with divisors {1, 6, 10, 15};
a(5) = 105 with divisors {1, 7, 15, 21, 35};
a(6) = 910 with divisors {1, 35, 65, 91, 130, 455};
a(7) = 561 with divisors {1, 3, 11, 17, 33, 51, 187};
a(8) = 1365 with divisors {1, 13, 21, 91, 105, 195, 273, 455};
a(9) = 5005 with divisors {1, 11, 55, 65, 77, 143, 385, 715, 1001};
a(10) = 5565 with divisors {1, 7, 15, 21, 35, 105, 265, 371, 1113, 1855};
a(11) = 11305 with divisors {1, 17, 19, 35, 85, 119, 323, 595, 665, 1615, 2261}.

Cf. A182816, A272538, A279024, A371883, A371884.

f[n_] := DivisorSum[n, 1 &, PowerMod[#, n, n] == # &]; seq[max_] := Module[{t = Table[0, {max}], c = 0, n = 1, i}, While[c < max, i = f[n] + 1; If[i <= max && t[[i]] == 0, c++; t[[i]] = n]; n++]; t]; seq[18] (* Amiram Eldar, Apr 11 2024 *)

from sympy import divisors
from itertools import count, islice
def f(n, divs): return sum(1 for d in divs if pow(d, n, n) == d%n)
def agen(verbose=False): # generator of terms
    adict, n = dict(), 0
    for k in count(1):
        divs = divisors(k)[1:]
        if len(divs) < n: continue
        v = f(k, divs)
        if v not in adict:
            adict[v] = k
            if verbose: print("FOUND", v, k)
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 15))) # Michael S. Branicky, Apr 10 2024, updated Apr 17 2024 after Jon E. Schoenfield

a(12)-a(25) from Michael S. Branicky, Apr 10 2024

a(26)-a(35) from Jon E. Schoenfield, 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 *)

A378387 a(n) is the number of proper divisors d of n such that (-d)^n == -d (mod n).

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Dec 23 2024

nonn

From Robert Israel, Dec 27 2024: (Start)
If n > 1 is odd, a(n) > 0 as d = 1 works.
a(n) = 1 if n is a prime power (A246655). (End)

			a(4) = 0 because the proper divisors of 4 are 1, 2 and
(-1)^4 (mod 4) is not congruent to 3 (mod 4);
(-2)^4 (mod 4) is not congruent to 2 (mod 4).
a(5) = 1 because the only proper divisor of 5 is 1 and
(-1)^5 (mod 5) == 4 (mod 5).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A032741, A371883.

[#[d: d in [1..n-1] | n mod d eq 0 and (-d)^n mod n eq n-d]: n in [1..100]];

f:= proc(n) nops(select((t -> (-t)&^n + t mod n = 0), numtheory:-divisors(n) minus {n})) end proc:
map(f, [$1..100]); # Robert Israel, Dec 27 2024

a[n_] := DivisorSum[n, 1 &, # < n && PowerMod[n - #, n, n] == n - # &]; Array[a, 100] (* Amiram Eldar, Dec 23 2024 *)

Edited by N. J. A. Sloane, Jan 11 2025

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.

Original entry on oeis.org

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, 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, 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

Offset: 1

Juri-Stepan Gerasimov, May 12 2024

nonn

			a(12) = 3: 1 divides 12, and 1^12 mod 12 = 1;
           2 divides 12, and 2^12 mod 12 = 4;
           3 divides 12, but 3^12 mod 12 = 9 (not a divisor of 12);
           4 divides 12, and 4^12 mod 12 = 4;
           6 divides 12, but 6^12 mod 12 = 0 (not a divisor of 12);
           12 divides 12, but 12^12 mod 12 = 0 (not a divisor of 12).

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

Cf. A371883.

[&+[#[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]];

a[n_] := DivisorSum[n, 1 &, (m = PowerMod[#, n, n]) > 0 && Divisible[n, m] &]; Array[a, 100] (* Amiram Eldar, May 13 2024 *)

A372772(n) = { my(k); sumdiv(n, d, k=lift(Mod(d^n,n)); k > 0 && 0==(n%k)); }; \\ Antti Karttunen, May 13 2024

from sympy import divisors
def a(n):
    divs = set(divisors(n)[:-1])
    return sum(1 for d in divs if pow(d, n, n) in divs)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, May 13 2024

A380656 a(n) is the number of divisors d such that -d^n mod n = d.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Feb 03 2025

nonn

Cf. A000005, A371883.

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

a[n_] := DivisorSum[n, 1 &, Mod[-PowerMod[#, n, n], n] == # &]; Array[a, 100] (* Amiram Eldar, Feb 04 2025 *)

A380858 a(n) is the number of primes p <= n such that p^(p + n) == p (mod p + n).

Original entry on oeis.org

0, 0, 2, 1, 2, 1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 2, 4, 1, 3, 1, 3, 1, 6, 0, 6, 1, 4, 2, 7, 1, 3, 0, 6, 3, 6, 1, 5, 2, 5, 2, 8, 1, 5, 1, 5, 1, 8, 0, 6, 2, 5, 1, 9, 0, 8, 1, 5, 3, 12, 1, 8, 1, 7, 2, 11, 1, 8, 2, 8, 2, 10, 1, 6, 0, 9, 1, 12, 1, 7, 1, 5, 1, 13, 0, 9, 3, 6, 1, 15

Offset: 1

Juri-Stepan Gerasimov, Feb 06 2025

			a(3) = 2 because 2^(2+3) = 32 mod (2+3) is equal to 2 and 3^(3+3) = 729 mod (3+3) is equal to 3;
a(4) = 1 because 2^(2+4) = 64 mod (2+4) is equal to 4, but not is equal to 2, and 3^(3+4) = 2187 mod (3+4) is equal to 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A371883.

[#[p: p in PrimesUpTo(n) | p^(p+n) mod (p+n) eq p]: n in [1..90]];

P:= NULL: R:= NULL:
for n from 1 to 100 do
  if isprime(n) then P:= P,n fi;
  R:= R, nops(select(p -> p &^ (p+n) mod (p+n) = p, [P]));
od:
R; # Robert Israel, Mar 12 2025

a[n_] := Count[Range[n], ?(PrimeQ[#] && PowerMod[#, # + n, # + n] == # &)]; Array[a, 100] (* _Amiram Eldar, Feb 06 2025 *)

a(n) = my(nb=0); forprime(p=2, n, if (Mod(p, p+n)^(p+n) == p, nb++)); nb; \\ Michel Marcus, Feb 06 2025

A380833 a(n) is the number of divisors d of n satisfying (-d)^n mod n = d.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Feb 06 2025

nonn

Cf. A000005, A371883, A380656.

[#[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, Feb 06 2025 *)

cp's OEIS Frontend

A371513 a(n) is the smallest number m with n divisors d such that d^m mod m = d.

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

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A378387 a(n) is the number of proper divisors d of n such that (-d)^n == -d (mod n).

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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.

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A380656 a(n) is the number of divisors d such that -d^n mod n = d.

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Crossrefs

Programs

Magma

Mathematica

A380858 a(n) is the number of primes p <= n such that p^(p + n) == p (mod p + n).

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Author

Keywords

Examples

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Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A380833 a(n) is the number of divisors d of n satisfying (-d)^n mod n = d.

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Author

Keywords

Crossrefs

Programs

Magma

Mathematica