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Search: a231409 -id:a231409
Displaying 1-10 of 10 results found. page 1
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A229303 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
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*#] == # &]
PROG
(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
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).
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Sep 21 2013
STATUS
approved
A229307 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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]
LINKS
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.
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[500], !g[2*#] == # &]
CROSSREFS
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).
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Sep 24 2013
STATUS
approved
A229300 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Complement of A229304.
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.)
LINKS
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
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], g[1806*#] == # &]
CROSSREFS
Cf. A031971, A231409.
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. 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. A229313 (numbers n such that A031971(47058*n) <> n (mod 47058*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).
Cf. A054377 (primary pseudoperfect numbers).
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Sep 19 2013
STATUS
approved
A229301 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.)
LINKS
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
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:
select(filter, [$1..100]); # Robert Israel, Dec 15 2020
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], g[42*#] == # &]
CROSSREFS
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).
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Sep 21 2013
STATUS
approved
A229302 Numbers n such that A031971(6*n) == n (mod 6*n). +0
17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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]
LINKS
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
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], g[6*#] == # &]
CROSSREFS
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).
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Sep 21 2013
STATUS
approved
A229304 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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]
LINKS
Table of n, a(n) for n=1..55.
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
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], !g[1806*#] == # &]
CROSSREFS
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).
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Sep 19 2013
STATUS
approved
A229305 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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]
LINKS
Table of n, a(n) for n=1..55.
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
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], !g[42*#] == # &]
CROSSREFS
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).
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Sep 23 2013
STATUS
approved
A229306 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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]
LINKS
Table of n, a(n) for n=1..57.
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
MATHEMATICA
g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], !g[6*#] == # &]
CROSSREFS
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).
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Sep 24 2013
STATUS
approved
A230311 Numbers n such that 1^(k*n) + 2^(k*n) + ... + (k*n)^(k*n) == k (mod k*n) for some k; that is, numbers n such that A031971(k*n) == k (mod k*n) for some k. +0
13
1, 2, 6, 42, 1806, 47058, 2214502422, 8490421583559688410706771261086 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Least such k is A231409. No other terms for n < 10^110 (see Grau, Oller-Marcen, Sondow (2015) p. 428). - Jonathan Sondow, Nov 30 2013
Same as quotients Q = m/n of solutions to the congruence 1^m + 2^m + . . . + m^m == n (mod m) with n|m. For Q > 1, a necessary condition is that Q be a primary pseudoperfect number A054377. The condition is not sufficient since the primary pseudoperfect number 52495396602 is not a member. - Jonathan Sondow, Jul 13 2014
LINKS
Table of n, a(n) for n=1..8.
Jose María Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^m + 2^m + . . . + m^m == n (mod m) with n|m, Monatshefte für Mathematik 177 (2015) 421-436, DOI 10.1007/s00605-014-0660-0
FORMULA
a(n) = A054377(n-1) for n = 2, 3, 4, 5, 6, 7, but a(8) = A054377(8). - Jonathan Sondow, Jul 13 2014
CROSSREFS
Cf. A031971, A231409.
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. 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. A229313 (numbers n such that A031971(47058*n) <> n (mod 47058*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).
Cf. A054377 (primary pseudoperfect numbers).
KEYWORD
nonn,more,hard
AUTHOR
José María Grau Ribas, Nov 06 2013
EXTENSIONS
Definition corrected by Jonathan Sondow, Nov 30 2013
STATUS
approved
A233045 1^m + 2^m + ... + m^m (mod m) for primary pseudoperfect numbers m. +0
2
1, 1, 1, 1, 5797, 272753965, 8749232767, 1045741078641946876220133713545 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
A031971(m) (mod m) for m in A054377 = 2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086. The known values of m for which 1^m + 2^m + ... + m^m == 1 (mod m) are m = 1, 2, 6, 42, 1806.
For any m and prime p | m, use Sum_{j=1..m} j^m == -m/p (mod p) if p-1 | m or == 0 (mod p) otherwise (see Lemma 3 in Grau et al.) and the Chinese Remainder Theorem.
LINKS
Table of n, a(n) for n=1..8.
J. M. Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n|m, arXiv:1309.7941 [math.NT].
FORMULA
a(n) = 1 for n = 1, 2, 3, 4.
EXAMPLE
The 1st primary pseudoperfect number is 2, and 1^2 + 2^2 = 5 == 1 (mod 2), so a(1) = 1.
MATHEMATICA
ps={2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086}; fa = FactorInteger; VonStaudt[n_] := Mod[n - Sum[If[IntegerQ[n/(fa[n][[i, 1]] - 1)], n/fa[n][[i, 1]], 0], {i, Length[fa[n]]}], n]; Table[VonStaudt[ps[[i]]], {i, 1, 8}]
CROSSREFS
Cf. A031971, A054377, A231409.
KEYWORD
more,nonn
AUTHOR
Jonathan Sondow, Dec 10 2013
STATUS
approved
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Last modified August 30 07:09 EDT 2024. Contains 375532 sequences. (Running on oeis4.)