Displaying 1-10 of 16 results found.
Numbers n such that A031971(1806*n) == n (mod 1806*n).
+0
17
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, 43, 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
COMMENTS
The asymptotic density is in [0.7747,0.812570].
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.)
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], g[1806*#] == # &]
CROSSREFS
Cf. A231562 (numbers n such that A031971(8490421583559688410706771261086*n) == n (mod 8490421583559688410706771261086*n)). Cf. A229312 (numbers n such that A031971(47058*n) == n (mod 47058*n)). Cf. A229313 (numbers n such that A031971(47058*n) <> n (mod 47058*n)). Cf. A054377 (primary pseudoperfect numbers).
Numbers n such that A031971(42*n) == n (mod 42*n).
+0
17
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
COMMENTS
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.)
MAPLE
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:
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], g[42*#] == # &]
Numbers m such that A031971(2*m) == m (mod 2*m).
+0
19
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
COMMENTS
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.
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:
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], g[2*#] == # &]
PROG
(PARI) b(n)=sum(k=1, n, Mod(k, n)^n);
for(n=1, 200, if(b(2*n)==n, print1(n, ", ")));
Numbers n such that A031971(1806*n) <> n (mod 1806*n).
+0
16
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
COMMENTS
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]
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], !g[1806*#] == # &]
Numbers n such that A031971(42*n) <> n (mod 42*n).
+0
16
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
COMMENTS
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]
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], !g[42*#] == # &]
Numbers n such that A031971(6*n) <> n (mod 6*n)
+0
16
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
COMMENTS
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]
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], !g[6*#] == # &]
Numbers k such that A031971(2*k) <> k (mod 2*k).
+0
19
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
COMMENTS
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]
LINKS
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.
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[500], !g[2*#] == # &]
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
MATHEMATICA
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*#] == # &]
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
MATHEMATICA
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*#] == # &]
7, 10, 26, 55, 57, 136, 155, 222, 253, 737, 876, 1027, 1081, 1552, 1711, 1751, 1962, 3197, 3403, 3775, 3916, 4401, 5671, 6176, 6567, 8251, 8515, 8702, 9316, 11026, 12195, 12742, 13301, 13861, 14878, 15657, 15931, 18145, 20242, 22387, 23126, 25651, 26202
MATHEMATICA
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[6*#] == # &]
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