A229303 Numbers m such that A031971(2*m) == m (mod 2*m).

1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 115, 116, 118, 119, 121, 122, 124, 125

Offset: 1

Author

José María Grau Ribas, Sep 21 2013

Keywords

• nonn

Comments

Complement of A229307.

The asymptotic density is in [0.583154, 0.58455].

The numbers k = 1, 2, 6, 42, 1806, 47058, 2214502422, 8490421583559688410706771261086 = A230311 are the only values of k such that the set {n: A031971(k*n) == n (mod k*n)} is nonempty. Its smallest element is n = 1, 1, 1, 1, 1, 5, 5, 39607528021345872635 = A231409. [Comment corrected and expanded by Jonathan Sondow, Dec 10 2013]

Up to (but excluding) the term 68 the exponents of even prime powers with squarefree neighbors. - Juri-Stepan Gerasimov, Apr 30 2016.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Jose María Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n

Crossrefs

Cf. A014117 (numbers k such that A031971(k)==1 (mod k)).

Cf. A229300 (numbers k such that A031971(1806*k)== k (mod 1806*k)).

Cf. A229301 (numbers k such that A031971(42*k) == k (mod 42*k)).

Cf. A229302 (numbers k such that A031971(6*k) == k (mod 6*k)).

Cf. A229303 (numbers k such that A031971(2*k) == k (mod 2*k)).

Cf. A229304 (numbers k such that A031971(1806*k) <> k (mod 1806*k)).

Cf. A229305 (numbers k such that A031971(42*k) <> k (mod 42*k)).

Cf. A229306 (numbers k such that A031971(6*k) <> k (mod 6*k)).

Cf. A229307 (numbers k such that A031971(2*k) <> k (mod 2*k)).

Cf. A229308 (primitive numbers in A229304).

Cf. A229309 (primitive numbers in A229305).

Cf. A229310 (primitive numbers in A229306).

Cf. A229311 (primitive numbers in A229307).

Programs

Maple

a:= proc(n) option remember; local m;
      for m from 1+`if`(n=1, 0, a(n-1)) do
        if (t-> m=(add(k&^t mod t, k=1..t) mod t))(2*m)
           then return m fi
      od
    end:
seq(a(n), n=1..200);  # Alois P. Heinz, May 01 2016

Mathematica

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], g[2*#] == # &]

PARI

b(n)=sum(k=1, n, Mod(k,n)^n);
for(n=1,200,if(b(2*n)==n,print1(n,", ")));
\\ Joerg Arndt, May 01 2016