cp's OEIS Frontend

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

%I A229303 #34 Jun 27 2024 22:21:25
%S A229303 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,
%T A229303 43,44,46,47,49,52,53,56,58,59,61,62,64,65,67,68,71,73,74,76,77,79,82,
%U A229303 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
%N A229303 Numbers m such that A031971(2*m) == m (mod 2*m).
%C A229303 Complement of A229307.
%C A229303 The asymptotic density is in [0.583154, 0.58455].
%C A229303 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]
%C A229303 Up to (but excluding) the term 68 the exponents of even prime powers with squarefree neighbors. - _Juri-Stepan Gerasimov_, Apr 30 2016.
%H A229303 Alois P. Heinz, <a href="/A229303/b229303.txt">Table of n, a(n) for n = 1..10000</a>
%H A229303 Jose María Grau, A. M. Oller-Marcen, and J. Sondow, <a href="http://arxiv.org/abs/1309.7941">On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n</a>
%p A229303 a:= proc(n) option remember; local m;
%p A229303       for m from 1+`if`(n=1, 0, a(n-1)) do
%p A229303         if (t-> m=(add(k&^t mod t, k=1..t) mod t))(2*m)
%p A229303            then return m fi
%p A229303       od
%p A229303     end:
%p A229303 seq(a(n), n=1..200);  # _Alois P. Heinz_, May 01 2016
%t A229303 g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], g[2*#] == # &]
%o A229303 (PARI) b(n)=sum(k=1, n, Mod(k,n)^n);
%o A229303 for(n=1,200,if(b(2*n)==n,print1(n,", ")));
%o A229303 \\ _Joerg Arndt_, May 01 2016
%Y A229303 Cf. A014117 (numbers k such that A031971(k)==1 (mod k)).
%Y A229303 Cf. A229300 (numbers k such that A031971(1806*k)== k (mod 1806*k)).
%Y A229303 Cf. A229301 (numbers k such that A031971(42*k) == k (mod 42*k)).
%Y A229303 Cf. A229302 (numbers k such that A031971(6*k) == k (mod 6*k)).
%Y A229303 Cf. A229303 (numbers k such that A031971(2*k) == k (mod 2*k)).
%Y A229303 Cf. A229304 (numbers k such that A031971(1806*k) <> k (mod 1806*k)).
%Y A229303 Cf. A229305 (numbers k such that A031971(42*k) <> k (mod 42*k)).
%Y A229303 Cf. A229306 (numbers k such that A031971(6*k) <> k (mod 6*k)).
%Y A229303 Cf. A229307 (numbers k such that A031971(2*k) <> k (mod 2*k)).
%Y A229303 Cf. A229308 (primitive numbers in A229304).
%Y A229303 Cf. A229309 (primitive numbers in A229305).
%Y A229303 Cf. A229310 (primitive numbers in A229306).
%Y A229303 Cf. A229311 (primitive numbers in A229307).
%K A229303 nonn
%O A229303 1,2
%A A229303 _José María Grau Ribas_, Sep 21 2013