A229300 -id:A229300 - cp's OEIS frontend

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

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

José María Grau Ribas, Sep 21 2013

nonn

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.

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

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).

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

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

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

A229307 Numbers k such that A031971(2k) <> k (mod 2k).

Original entry on oeis.org

3, 6, 9, 10, 12, 15, 18, 20, 21, 24, 27, 30, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 55, 57, 60, 63, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 90, 93, 96, 99, 100, 102, 105, 108, 110, 111, 114, 117, 120, 123, 126, 129, 130, 132, 135, 136, 138, 140, 141, 144

Offset: 1

José María Grau Ribas, Sep 24 2013

nonn

Complement of A229303.
The asymptotic density is in [0.41545, 0.416846].
If n is in A then k*n is in A for all natural number k.
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]

Amiram Eldar, Table of n, a(n) for n = 1..10000
José María Grau, Antonio M. Oller-Marcén and Jonathan Sondow, On the congruence 1^m + 2^m + ... + m^m = n (mod m), with n|m, Monatshefte für Mathematik, Vol. 177, No. 3 (2015), pp. 421-436, preprint, arXiv:1309.7941 [math.NT], 2013-2014.

Cf. A031971.
Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[500], !g[2*#] == # &]

A229301 Numbers n such that A031971(42n) == n (mod 42n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 44, 45, 46, 47, 48, 49, 51, 53, 54, 56, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82

Offset: 1

José María Grau Ribas, Sep 21 2013

nonn

Complement of A229305.
The asymptotic density is in [0.7880, 0.8079].
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.)

Robert Israel, Table of n, a(n) for n = 1..1000
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

Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

filter:= proc(n) local t,k;
  t:= add(k &^ (42*n) mod (42*n),k=1..42*n);
  t mod (42*n) = n
end proc:
select(filter, [$1..100]); # Robert Israel, Dec 15 2020

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n];Select[Range[100], g[42*#] == # &]

A229302 Numbers n such that A031971(6n) == n (mod 6n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 22, 23, 24, 25, 27, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 51, 53, 54, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 93

Offset: 1

José María Grau Ribas, Sep 21 2013

nonn

Complement of A229306.
The asymptotic density is in [0.6986, 0.7073].
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]

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

Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], g[6*#] == # &]

A229311 Primitive numbers in A229307.

Original entry on oeis.org

3, 10, 55, 136, 253, 406, 1081, 1378, 1711, 2485, 3403, 3916, 5671, 6328, 8515, 9316, 11026, 13861, 14878, 15931, 18145, 19306, 25651, 27028, 28441, 31375, 32896, 34453, 36046, 42778, 48205, 50086, 60031, 62128, 64261, 73153, 75466, 87571, 92665, 97903

Offset: 1

José María Grau Ribas, Sep 24 2013

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..67
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

Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, 1, n}], n]; tachar[lis_, num_] := Select[lis, ! IntegerQ[#1/num] &]; primi[{}]={}; primi[lis_] := Join[{lis[[1]]}, primi[tachar[lis, lis[[1]]]]]; primi@Select[Range[500], !g[2*#] == # &]

A229304 Numbers n such that A031971(1806n) <> n (mod 1806n).

Original entry on oeis.org

10, 20, 26, 30, 40, 50, 52, 55, 57, 58, 60, 70, 78, 80, 90, 100, 104, 110, 114, 116, 120, 130, 136, 140, 150, 155, 156, 160, 165, 170, 171, 174, 180, 182, 190, 200, 208, 210, 220, 222, 228, 230, 232, 234, 240, 250, 253, 260, 270, 272, 275, 280, 285, 286, 290

Offset: 1

José María Grau Ribas, Sep 19 2013

nonn

Complement of A229300.
The asymptotic density is in [0.1921, 0.212].
If n is in A then k*n is in A for all natural number k.
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]

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

Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], !g[1806*#] == # &]

A229305 Numbers n such that A031971(42n) <> n (mod 42n).

Original entry on oeis.org

10, 20, 26, 30, 40, 43, 50, 52, 55, 57, 58, 60, 70, 78, 80, 86, 90, 100, 104, 110, 114, 116, 120, 129, 130, 136, 140, 150, 155, 156, 160, 165, 170, 171, 172, 174, 180, 182, 190, 200, 208, 210, 215, 220, 222, 228, 230, 232, 234, 240, 250, 253, 258, 260, 270

Offset: 1

José María Grau Ribas, Sep 23 2013

nonn

Complement of A229301.
The asymptotic density is in [0.2091, 0.2317].
If n is in A then k*n is in A for all natural number k.
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]

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

Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], !g[42*#] == # &]

A229306 Numbers n such that A031971(6n) <> n (mod 6n).

Original entry on oeis.org

7, 10, 14, 20, 21, 26, 28, 30, 35, 40, 42, 49, 50, 52, 55, 56, 57, 60, 63, 70, 77, 78, 80, 84, 90, 91, 98, 100, 104, 105, 110, 112, 114, 119, 120, 126, 130, 133, 136, 140, 147, 150, 154, 155, 156, 160, 161, 165, 168, 170, 171, 175, 180, 182, 189, 190, 196

Offset: 1

José María Grau Ribas, Sep 24 2013

nonn

Complement of A229302.
The asymptotic density is in [0.2927, 0.3014].
If n is in A then k*n is in A for all natural number k.
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]

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

Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], !g[6*#] == # &]

A229308 Primitive numbers in A229304.

Original entry on oeis.org

10, 26, 55, 57, 58, 136, 155, 222, 253, 346, 355, 381, 737, 876, 904, 1027, 1055, 1081, 1552, 1711, 1751, 1962, 2155, 2696, 2758, 3197, 3403, 3411, 3775, 3916, 4063, 4132, 4401, 5093, 5671, 6176, 6455, 6567, 7111, 7226, 8251, 8515, 8702, 9294, 9316, 9465

Offset: 1

José María Grau Ribas, Sep 20 2013

nonn

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

Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, 1, n}], n]; tachar[lis_, num_] := Select[lis, ! IntegerQ[#1/num] &];primi[{}] = {}; primi[lis_] := Join[{lis[[1]]}, primi[tachar[lis, lis[[1]]]]]; primi@Select[Range[70], ! g[1806*#] == # &]

A229309 Primitive numbers in A229305.

Original entry on oeis.org

10, 26, 43, 55, 57, 58, 136, 155, 222, 253, 355, 381, 737, 876, 904, 1027, 1055, 1081, 1552, 1711, 1751, 1962, 2696, 2758, 3197, 3403, 3411, 3775, 3916, 4063, 4401, 5093, 5671, 6176, 6567, 7111, 8251, 8515, 8702, 9316, 9465, 10768, 11026, 12195, 12742, 13301

Offset: 1

José María Grau Ribas, Sep 24 2013

nonn

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

Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, 1, n}], n]; tachar[lis_, num_] := Select[lis, ! IntegerQ[#1/num] &]; primi[{}]={}; primi[lis_] := Join[{lis[[1]]}, primi[tachar[lis, lis[[1]]]]]; primi@Select[Range[80], !g[42*#] == # &]

cp's OEIS Frontend

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

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

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A229301 Numbers n such that A031971(42*n) == n (mod 42*n).

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Maple

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A229302 Numbers n such that A031971(6*n) == n (mod 6*n).

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Mathematica

A229311 Primitive numbers in A229307.

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Mathematica

A229304 Numbers n such that A031971(1806*n) <> n (mod 1806*n).

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A229305 Numbers n such that A031971(42*n) <> n (mod 42*n).

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A229306 Numbers n such that A031971(6*n) <> n (mod 6*n).

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A229308 Primitive numbers in A229304.

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

A229307 Numbers k such that A031971(2k) <> k (mod 2k).

A229301 Numbers n such that A031971(42n) == n (mod 42n).

A229302 Numbers n such that A031971(6n) == n (mod 6n).

A229304 Numbers n such that A031971(1806n) <> n (mod 1806n).

A229305 Numbers n such that A031971(42n) <> n (mod 42n).

A229306 Numbers n such that A031971(6n) <> n (mod 6n).