A230311 -id:A230311 - cp's OEIS frontend

A231409 Least k with 1^(km) + 2^(km) + ... + (km)^(km) == k (mod k*m) for m in A230311.

1, 1, 1, 1, 1, 5, 5, 39607528021345872635

Offset: 1

Jonathan Sondow, Nov 30 2013

Least k with A031971(k*m) == k (mod k*m) for m in A230311.

See A031971 and A230311 for more comments and crossrefs.

			1^m + 2^m + ... + m^m == 1 (mod m) for the first 5 terms m = 1, 2, 6, 42, 1806 of A230311, so a(n) = 1 for n <= 5.

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].

Cf. A031971, A229300, A229301, A229302, A229303, A230311.

a(2) = A229303(1), a(3) = A229302(1), a(4) = A229301(1), a(5) = A229300, a(6) = A229312(1).

A054377 Primary pseudoperfect numbers: numbers n > 1 such that 1/n + sum 1/p = 1, where the sum is over the primes p | n.

Original entry on oeis.org

2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086

Offset: 1

Eric W. Weisstein

Primary pseudoperfect numbers are the solutions of the "differential equation" n' = n-1, where n' is the arithmetic derivative of n. - Paolo P. Lava, Nov 16 2009
Same as n > 1 such that 1 + sum n/p = n (and the only known numbers n > 1 satisfying the weaker condition that 1 + sum n/p is divisible by n). Hence a(n) is squarefree, and is pseudoperfect if n > 1. Remarkably, a(n) has exactly n (distinct) prime factors for n < 9. - Jonathan Sondow, Apr 21 2013
From the Wikipedia article: it is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any odd primary pseudoperfect numbers. - Daniel Forgues, May 27 2013
Since the arithmetic derivative of a prime p is p' = 1, 2 is obviously the only prime in the sequence. - Daniel Forgues, May 29 2013
Just as 1 is not a prime number, 1 is also not a primary pseudoperfect number, according to the original definition by Butske, Jaje, and Mayernik, as well as Wikipedia and MathWorld. - Jonathan Sondow, Dec 01 2013
Is it always true that if a primary pseudoperfect number N > 2 is adjacent to a prime N-1 or N+1, then in fact N lies between twin primes N-1, N+1? See A235139. - Jonathan Sondow, Jan 05 2014
Also, integers n > 1 such that A069359(n) = n - 1. - Jonathan Sondow, Apr 16 2014

			From _Daniel Forgues_, May 24 2013: (Start)
With a(1) = 2, we have 1/2 + 1/2 = (1 + 1)/2 = 1;
with a(2) = 6 = 2 * 3, we have
  1/2 + 1/3 + 1/6 = (3 + 2 + 1)/6 = (1*3 + 3)/(2*3) = (1 + 1)/2 = 1;
with a(3) = 42 = 6 * 7, we have
  1/2 + 1/3 + 1/7 + 1/42 = (21 + 14 + 6 + 1)/42 =
  (3*7 + 2*7 + 7)/(6*7) = (3 + 2 + 1)/6 = 1;
with a(4) = 1806 = 42 * 43, we have
  1/2 + 1/3 + 1/7 + 1/43 + 1/1806 = (903 + 602 + 258 + 42 + 1)/1806 =
  (21*43 + 14*43 + 6*43 + 43)/(42*43) = (21 + 14 + 6 + 1)/42 = 1;
with a(5) = 47058 (not oblong number), we have
  1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/47058 =
  (23529 + 15686 + 4278 + 2046 + 1518 + 1)/47058 = 1.
For n = 1 to 8, a(n) has n prime factors:
  a(1) = 2
  a(2) = 2 * 3
  a(3) = 2 * 3 *  7
  a(4) = 2 * 3 *  7 * 43
  a(5) = 2 * 3 * 11 * 23 *  31
  a(6) = 2 * 3 * 11 * 23 *  31 * 47059
  a(7) = 2 * 3 * 11 * 17 * 101 *   149 *       3109
  a(8) = 2 * 3 * 11 * 23 *  31 * 47059 * 2217342227 * 1729101023519
If a(n)+1 is prime, then a(n)*[a(n)+1] is also primary pseudoperfect. We have the chains: a(1) -> a(2) -> a(3) -> a(4); a(5) -> a(6). (End)
A primary pseudoperfect number (greater than 2) is oblong if and only if it is not the initial member of a chain. - _Daniel Forgues_, May 29 2013
If a(n)-1 is prime, then a(n)*(a(n)-1) is a Giuga number (A007850). This occurs for a(2), a(3), and a(5). See A235139 and the link "The p-adic order . . .", Theorem 8 and Example 1. - _Jonathan Sondow_, Jan 06 2014

M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT]
W. Butske, L. M. Jaje, and D. R. Mayernik, On the Equation Sum_{p|N} 1/p + 1/N = 1, Pseudoperfect numbers and partially weighted graphs, Math. Comput., 69 (1999), 407-420. [Title corrected by _Jonathan Sondow_, Apr 11 2012]
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], 2013.
J. M. Grau, A. M. Oller-Marcén, and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.
J. Sondow and K. MacMillan, Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2, Integers 11 (2011), #A34.
J. Sondow and K. MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation, Amer. Math. Monthly, 124 (2017) 232-240; arXiv:math/1812.06566 [math.NT], 2018.
J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
Eric Weisstein's World of Mathematics, Primary pseudoperfect number.
Wikipedia, Primary pseudoperfect number.
OEIS Wiki, Primary pseudoperfect numbers.

Cf. A005835, A007850, A069359, A168036, A190272, A191975, A203618, A216825, A216826, A230311, A235137, A235138, A235139, A236433.

pQ[n_] := (f = FactorInteger[n]; 1/n + Sum[1/f[[i]][[1]], {i, Length[f]}] == 1)
Select[Range[2, 10^6], pQ[#] &] (* Robert Price, Mar 14 2020 *)

isok(n) = if (n > 1, my(f=factor(n)[,1]); 1/n + sum(k=1, #f, 1/f[k]) == 1); \\ Michel Marcus, Oct 05 2017

from sympy import primefactors
A054377 = [n for n in range(2,10**5) if sum([n/p for p in primefactors(n)]) +1 == n] # Chai Wah Wu, Aug 20 2014

A031971(a(n)) (mod a(n)) = A233045(n). - Jonathan Sondow, Dec 11 2013
A069359(a(n)) = a(n) - 1. - Jonathan Sondow, Apr 16 2014
a(n) == 36*(n-2) + 6 (mod 288) for n = 2,3,..,8. - Kieren MacMillan and Jonathan Sondow, Sep 20 2017

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

Original entry on oeis.org

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*#] == # &]

A229300 Numbers n such that A031971(1806n) == n (mod 1806n).

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, 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

Offset: 1

José María Grau Ribas, Sep 19 2013

nonn

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.)

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. 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).

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

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*#] == # &]

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*#] == # &]

cp's OEIS Frontend

A231409 Least k with 1^(k*m) + 2^(k*m) + ... + (k*m)^(k*m) == k (mod k*m) for m in A230311.

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A054377 Primary pseudoperfect numbers: numbers n > 1 such that 1/n + sum 1/p = 1, where the sum is over the primes p | n.

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

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Maple

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

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A229300 Numbers n such that A031971(1806*n) == n (mod 1806*n).

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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

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

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A231409 Least k with 1^(km) + 2^(km) + ... + (km)^(km) == k (mod k*m) for m in A230311.

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

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

A229300 Numbers n such that A031971(1806n) == n (mod 1806n).

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).