A106741 -id:A106741 - cp's OEIS frontend

A014117 Numbers n such that m^(n+1) == m (mod n) holds for all m.

1, 2, 6, 42, 1806

Offset: 1

"Somebody incorrectly remembered Fermat's little theorem as saying that the congruence a^{n+1} = a (mod n) holds for all a if n is prime" (Zagier). The sequence gives the set of integers n for which this property is in fact true.
If i == j (mod n), then m^i == m^j (mod n) for all m. The latter congruence generally holds for any (m, n)=1 with i == j (mod k), k being the order of m modulo n, i.e., the least power k for which m^k == 1 (mod n). - Lekraj Beedassy, Jul 04 2002
Also, numbers n such that n divides denominator of the n-th Bernoulli number B(n) (cf. A106741). Also, numbers n such that 1^n + 2^n + 3^n + ... + n^n == 1 (mod n). Equivalently, numbers n such that B(n)*n == 1 (mod n). Equivalently, Sum_{prime p, (p-1) divides n} n/p == -1 (mod n). It is easy to see that for n > 1, n must be an even squarefree number. Moreover, the set P of prime divisors of all such n satisfies the property: if p is in P, then p-1 is the product of distinct elements of P. This set is P = {2, 3, 7, 43}, implying that the sequence is finite and complete. - Max Alekseyev, Aug 25 2013
In 2005, B. C. Kellner proved E. W. Weisstein's conjecture that denom(B_n) = n only if n = 1806. - Jonathan Sondow, Oct 14 2013
Squarefree numbers n such that b^n == 1 (mod n^2) for every b coprime to n. Squarefree terms of A341858. - Thomas Ordowski, Aug 05 2024
Conjecture: Numbers n such that gcd(d+1, n) > 1 for every proper divisor d of n. Verified up to 10^696. - David Radcliffe, May 29 2025

M. A. Alekseyev, J. M. Grau, and A. M. Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT], 2016-2018.
John H. Castillo and Jhony Fernando Caranguay Mainguez, The set of k-units modulo n, arXiv:1708.06812 [math.NT], 2017.
Yongyi Chen and Tae Kyu Kim, On Generalized Carmichael Numbers, arXiv:2103.04883 [math.NT], 2021.
J. Dyer-Bennet, A Theorem in Partitions of the Set of Positive Integers, Amer. Math. Monthly, 47(1940) pp. 152-4.
L. Halbeisen and N. Hungerbühler, On generalised Carmichael numbers, Hardy-Ramanujan Society, 1999, 22 (2), pp.8-22. (hal-01109575)
J. M. Grau, A. M. Oller-Marcen, and Jonathan Sondow, On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n|m, arXiv 1309.7941 [math.NT], 2013; Monatsh. Math., 177 (2015), 421-436.
B. C. Kellner, The equation denom(B_n) = n has only one solution, preprint 2005.
Don Reble, A014117 and related OEIS sequences (shows there are no other terms)
Jonathan Sondow and K. MacMillan, Reducing the Erdos-Moser equation 1^n + 2^n + … + k^n = (k+1)^n modulo k and k^2, arXiv:1011.2154 [math.NT], 2010; see Prop. 2; Integers, 11 (2011), article A34.
Eric Weisstein's World of Mathematics, Bernoulli Number
D. Zagier, Problems posed at the St Andrews Colloquium, 1996

Squarefree terms of A124240. - Robert Israel and Thomas Ordowski, Jun 23 2017

Cf. A007018, A031971, A054377, A100016, A242927, A341858.

r[n_] := Reduce[ Mod[m^(n+1) - m, n] == 0, m, Integers]; ok[n_] := Range[n]-1 === Simplify[ Mod[ Flatten[ m /. {ToRules[ r[n][[2]] ]}], n], Element[C[1], Integers]]; ok[1] = True; A014117 = {}; Do[ If[ok[n], Print[n]; AppendTo[ A014117, n] ], {n, 1, 2000}] (* Jean-François Alcover, Dec 21 2011 *)
Select[Range@ 2000, Function[n, Times @@ Boole@ Map[Function[m, PowerMod[m, n + 1, n] == Mod[m, n]], Range@ n] > 0]] (* Michael De Vlieger, Dec 30 2016 *)

[n for n in range(1, 2000) if all(pow(m, n+1, n) == m for m in range(n))] # David Radcliffe, May 29 2025

For n <= 5, a(n) = a(n-1)^2 + a(n-1) with a(0) = 1. - Raphie Frank, Nov 12 2012

a(n+1) = A007018(n) = A054377(n) = A100016(n) for n = 1, 2, 3, 4. - Jonathan Sondow, Oct 01 2013

A281662 (Denominator of Bernoulli(2*n)) read mod n.

Original entry on oeis.org

0, 0, 0, 2, 1, 0, 6, 6, 6, 0, 6, 6, 6, 2, 12, 14, 6, 12, 6, 10, 0, 8, 6, 18, 16, 4, 15, 2, 6, 0, 6, 30, 9, 30, 31, 30, 6, 30, 3, 10, 6, 0, 6, 30, 3, 30, 6, 42, 6, 30, 42, 30, 6, 30, 33, 6, 42, 30, 6, 30, 6, 30, 42, 62, 1, 24, 6, 30, 42, 50, 6, 6, 6, 30, 72, 30, 61

Offset: 1

Seiichi Manyama, Jan 26 2017

nonn

Odd terms at: 5, 27, 33, 35, 39, 45, 55, 65, 77, 81, 99, 105, 121, etc. - Robert G. Wilson v, Jan 26 2017

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A002445, A106741, A281648.

f[n_] := Mod[Denominator[BernoulliB[2 n]], n]; Array[f, 77] (* Robert G. Wilson v, Jan 26 2017 *)

a(n) = denominator(bernfrac(2*n)) % n; \\ Michel Marcus, Jan 29 2017

def bernoulli(n)
  ary = []
  a = []
  (0..n).each{|i|
    a << 1r / (i + 1)
    i.downto(1){|j| a[j - 1] = j * (a[j - 1] - a[j])}
    ary << a[0]
  }
  ary
end
def A281662(n)
  a = bernoulli(2 * n)
  (1..n).map{|i| a[2 * i].denominator % i}
end

a(n) = A002445(n) mod n.

A099008 Numbers n such that the denominator of the 2n-th Bernoulli number is divisible by n but sum_{d|n} sigma(d)/phi(d) is not an integer.

Original entry on oeis.org

546, 903, 1806, 2730, 6162, 6510, 9030, 10230, 12090, 12246, 12810, 15834, 20130, 20670, 23478, 23790, 28938, 30030, 30810, 43134, 44310, 52374, 56730, 61230, 71610, 79170, 84630, 85722, 88410, 99330, 109230, 117390, 132990, 140910, 144690, 154770, 164010

Offset: 1

Benoit Cloitre, Nov 07 2004

nonn

Intersection of A106741 and complement of A068991. - Michel Marcus, Dec 07 2013

			The denominator of the 1806th Bernoulli number is 1806 and Sum_{d|1806} sigma(d)/phi(d) = 172/3 is not an integer.

Cf. A068991.

lista() = {v = readvec("b106741.txt"); for (i=1, #v, vi = v[i]; if (denominator(sumdiv(vi, d, sigma(d)/eulerphi(d))) != 1, print1(vi, ", ")));} \\ Michel Marcus, Dec 07 2013

Extended using b-file from A106741 by Michel Marcus, Dec 07 2013

A212197 Numbers k that divide the 3k-th Clausen number.

Original entry on oeis.org

1, 2, 6, 14, 42, 114, 602, 798, 1806, 5334, 34314, 101346, 229362, 4357878, 9786714, 12198858, 168241542, 185947566, 231778302, 524550894

Offset: 1

Peter Luschny, May 05 2012

The classical Clausen numbers are given in A141056. See A160014 for generalizations. Related sequences are A014117 and A106741.

Thomas Clausen, Theorem. Lehrsatz aus einer Abhandlung ueber die Bernoullischen Zahlen, Astr. Nachr. 17 (22) (1840), 351-352.

Cf. A014117, A027760, A106741, A141056, A160014.

(* This program is not convenient for more than 15 terms *) c[n_] := Sum[Boole[PrimeQ[d+1]]/(d+1), {d, Divisors[n]}] // Denominator; Reap[For[n = 1, n < 10^7, n++, If[Divisible[c[3*n], n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, May 21 2013 *)

A212197_list(searchlimit) =
{
    for (n=1, searchlimit,
        p = 1;
        fordiv(3*n, d,
            r = d + 1;
            if (isprime(r), p = p*r;)
        );
        if (Mod(p, n) == 0, print1(n, ", "));
    );
}

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A014117 Numbers n such that m^(n+1) == m (mod n) holds for all m.

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A281662 (Denominator of Bernoulli(2*n)) read mod n.

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A099008 Numbers n such that the denominator of the 2n-th Bernoulli number is divisible by n but sum_{d|n} sigma(d)/phi(d) is not an integer.

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A212197 Numbers k that divide the 3k-th Clausen number.

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