A279024 -id:A279024 - 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

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

Original entry on oeis.org

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 *)

A278921 Semiprimes of the form p*q where p < q such that q divides p^(q+1) + 1 and (q-p)^(q+1) + 1.

Original entry on oeis.org

10, 15, 65, 221, 493, 671, 1147, 1219, 3439, 5069, 12209, 14893, 20737, 24503, 30083, 49813, 61937, 77507, 91277, 97297, 100337, 102719, 109283, 109783, 113521, 132427, 144301, 178991, 204851, 244523, 245041, 246559, 257149, 258749, 312167, 339497, 397219, 433091, 434617, 461893, 465763

Offset: 1

Juri-Stepan Gerasimov, Dec 01 2016

nonn

q is always a Pythagorean prime (A002144).
Semiprimes of the form p*q where p < q such that q divides p^(q+1) + k and (q-p)^(q+1) + k:
k = 1: (this sequence);
k = 2: 6, 33, 119, 247, 451, ...
k = 3: 14, 35, 91, 341, ...
k = 4: 39, 145, 371, ...
For every positive odd number q (whether prime or not), every integer p in 0..q, and every integer k, if q divides p^(q+1) + k, then it necessarily follows that q also divides (q-p)^(q+1) + k; thus, this sequence could be more simply defined as "Semiprimes of the form p*q where p < q such that q divides p^(q+1) + 1." - Jon E. Schoenfield, Dec 07 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A002144, A006881, A279024.

Take[#, 41] &@ Union@ Flatten@ Table[Function[q, q Select[Prime@ Range@ n, Function[p, And[Divisible[p^(q + 1) + 1, q], Divisible[(q - p)^(q + 1) + 1, q]]]]]@ Prime@ n, {n, 600}] (* Michael De Vlieger, Dec 02 2016 *)

list(lim)=my(v=List()); forprime(q=5,lim\2, if(q%4>2, next); forprime(p=2,min(lim\q,q-2), if(Mod(p,q)^(q+1)==-1 && Mod(q-p,q)^(q+1)==-1, listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Dec 02 2016

A279680 Definition: m < n is an extradivisor of n if for some positive k < n, m | n | k^(n+1) + m and n | (n-k)^(n+1) + m. This sequence gives the smallest number with n extradivisors.

Original entry on oeis.org

1, 2, 45, 105, 1365, 1305, 4305, 11445

Offset: 0

Juri-Stepan Gerasimov, Dec 16 2016

			a(0) = 1 with extradivisors {};
a(1) = 2 with extradivisor {1};
a(2) = 45 with extradivisors {5, 9};
a(3) = 105 with extradivisors {5, 21, 35};
a(4) = 1365 with extradivisors {35, 105, 195, 455};
a(5) = 1305 with extradivisors {5, 9, 29, 45, 261}.

Cf. A272538, A279024.

First /@ Values@ KeySort@ PositionIndex@ Table[Count[DeleteCases[Most@ Divisors@ n, d_ /; EvenQ@ d], m_ /; Total@ Boole@ Map[Function[k, And[Mod[PowerMod[k, (n + 1), n] + m, n] == 0, Mod[PowerMod[(n - k), (n + 1), n] + m, n] == 0]], Range[n - 1]] > 0], {n, 1500}] (* Michael De Vlieger, Dec 17 2016, Version 10 *)

a(3)-a(7) from Michael De Vlieger, Dec 07 2016

Definition edited by N. J. A. Sloane, Jun 19 2020

cp's OEIS Frontend

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

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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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A278921 Semiprimes of the form p*q where p < q such that q divides p^(q+1) + 1 and (q-p)^(q+1) + 1.

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A279680 Definition: m < n is an extradivisor of n if for some positive k < n, m | n | k^(n+1) + m and n | (n-k)^(n+1) + m. This sequence gives the smallest number with n extradivisors.

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