A229311 -id:A229311 - cp's OEIS frontend

A229310 Primitive numbers in A229306.

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

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[500], !g[6*#] == # &]

A230311 Numbers n such that 1^(kn) + 2^(kn) + ... + (kn)^(kn) == k (mod kn) for some k; that is, numbers n such that A031971(kn) == k (mod k*n) for some k.

Original entry on oeis.org

1, 2, 6, 42, 1806, 47058, 2214502422, 8490421583559688410706771261086

Offset: 1

José María Grau Ribas, Nov 06 2013

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

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

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).

a(n) = A054377(n-1) for n = 2, 3, 4, 5, 6, 7, but a(8) = A054377(8). - Jonathan Sondow, Jul 13 2014

Definition corrected by Jonathan Sondow, Nov 30 2013

A229312 Numbers n such that A031971(47058n) == n (mod 47058n).

Original entry on oeis.org

5, 15, 25, 45, 55, 65, 75, 85, 95, 115, 125, 135, 145, 155, 165, 185, 195, 205, 215, 225, 255, 265, 275, 295, 305, 325, 345, 355, 365, 375, 395, 405, 415, 425, 435, 445, 465, 475, 485, 495, 505, 515, 535, 545, 555, 565, 575, 585, 605, 615, 625, 635, 645, 655

Offset: 1

José María Grau Ribas, Oct 20 2013

nonn

The number 47058 occurring in the name is the sixth term of A230311.
The asymptotic density lies in the interval [0.0560465, 0.0800567].
Complement of A230313 .
For n<235295, A031971(47058*n) == n (mod 47058*n) if and only if A031971(2214502422*n) <> n (mod 2214502422*n).
The numbers in A230311 are the values of k such that the set {n : A031971(k*n)== n (mod k*n)} is nonempty.

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, A230311.
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).

fa = FactorInteger; Car[k_, n_] := Mod[n - Sum[If[IntegerQ[k/(fa[n][[i,
   1]] - 1)], n/fa[n][[i, 1]], 0], {i, 1, Length[fa[n]]}], n]; supercar[k_, n_] := If[k == 1 || Mod[k, 2] == 0 || Mod[n, 4] > 0, Car[k,n], Mod[Car[k, n] - n/2, n]]; Select[Range[1000], supercar[47058*#, 47058*#] == # &]

A231562 Numbers n such that A031971(8490421583559688410706771261086n) == n (mod 8490421583559688410706771261086n).

Original entry on oeis.org

39607528021345872635, 118822584064037617905, 198037640106729363175, 356467752192112853715, 435682808234804598985, 514897864277496344255, 594112920320188089525, 673327976362879834795, 752543032405571580065, 910973144490955070605, 990188200533646815875

Offset: 1

José María Grau Ribas, Nov 16 2013

nonn

The number 8490421583559688410706771261086 occurring in the name is the 8th term of A230311.

The numbers in A230311 are the values of k such that the set {n : A031971(k*n)== n (mod k*n)} is nonempty.

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, A230311.
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).

fa = FactorInteger; Car[k_, n_] := Mod[n - Sum[If[IntegerQ[k/(fa[n][[i, 1]] - 1)], n/fa[n][[i,1]], 0], {i, 1, Length[fa[n]]}], n]; supercar[k_, n_] := If[k == 1 ||  Mod[k, 2] == 0 || Mod[n, 4] > 0, Car[k, n], Mod[Car[k, n] - n/2,]];  Select[39607528021345872635*Range[15],supercar[8490421583559688410706771261086*#, 8490421583559688410706771261086*#] == # &]

A182398 a(n) = (Sum_{k=1..2n} k^2n) mod 2n.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 7, 8, 3, 6, 11, 4, 13, 14, 5, 16, 17, 6, 19, 12, 1, 22, 23, 8, 25, 26, 9, 28, 29, 58, 31, 32, 11, 34, 35, 12, 37, 38, 13, 24, 41, 2, 43, 44, 15, 46, 47, 16, 49, 30, 17, 52, 53, 18, 45, 56, 19, 58, 59, 116, 61, 62, 3, 64, 65, 22, 67, 68, 23

Offset: 1

Michel Lagneau, Apr 27 2012

nonn

Sum_{k=1..n} k^n (mod n) = 0 if n odd.
Properties of this sequence:
a(n) = 1 for n = 1, 3, 21, 903, ...
a(n) = n if n not divisible by 3;
a(3*n) = n except for n = 7, 10, 14, 20, 21, 26, 28, 30, 35, ...
a(21*n) = n, except for n = 10, 20, 26, 30, 40, 43, 50, 52, ...
a(903*n) = n, except for n = 10, ....
It appears that a(A007018(n)/2) = 1 and conjecturally a(m*A007018(n)/2) = m for a majority of value m.
No, a(A007018(n)/2) <> 1 for n > 4. (For example, a(A007018(5)/2) = a(1631721) = 1807.) - Jonathan Sondow, Oct 18 2013
0 < a(n) < 10 for n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 21, 24, 27, 42, 63, 84, 105, 126, 147, 168, 189, 903, 1806, 2709, 3612, 4515, 5418, 6321, 7224, 8127, .... Search limit was 25000. - Robert G. Wilson v, Jun 18 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..2499 from Michel Lagneau)
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, 2013.
Mathematics Stack Exchange, 1^n + 2^n + ... + (p-1)^n mod p = ?

Cf. A007018, A014117, A031971, A229307, A229311.

for n from 1 to 100 do: s:=sum('k^(2*n)', 'k'=1..2*n)
: x:=irem(s,2*n): printf(`%d, `,x):od:
# second Maple program:
a:= n-> add(k&^(2*n) mod (2*n), k=1..2*n) mod (2*n):
seq(a(n), n=1..100);

Table[Mod[Total[PowerMod[Range[2*n], 2*n, 2*n]], 2*n], {n, 100}] (* T. D. Noe, Apr 28 2012 *)

a(n) = A031971(2n) mod 2n. - Jonathan Sondow, Oct 18 2013

A230313 Numbers n such that A031971(47058n) <> n (mod 47058n).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

José María Grau Ribas, Nov 16 2013

nonn

The asymptotic density lies in the interval [0.919943, 0.943954].
Complement of A229312.
The numbers in A230311 are the values of k such that the set {n : A031971(k*n)== n (mod k*n)} is nonempty.

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, A230311.
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).

fa = FactorInteger; Car[k_, n_] := Mod[n - Sum[If[IntegerQ[k/(fa[n][[i,
   1]] - 1)], n/fa[n][[i, 1]], 0], {i, 1, Length[fa[n]]}], n]; supercar[k_, n_] := If[k == 1 || Mod[k, 2] == 0 || Mod[n, 4] > 0, Car[k,n], Mod[Car[k, n] - n/2, n]]; Select[Range[1000], !supercar[47058*#, 47058*#] == # &]

cp's OEIS Frontend

A229310 Primitive numbers in A229306.

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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.

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A229312 Numbers n such that A031971(47058*n) == n (mod 47058*n).

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A231562 Numbers n such that A031971(8490421583559688410706771261086*n) == n (mod 8490421583559688410706771261086*n).

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A182398 a(n) = (Sum_{k=1..2n} k^2n) mod 2n.

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Maple

Mathematica

Formula

A230313 Numbers n such that A031971(47058*n) <> n (mod 47058*n).

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Author

Keywords

Comments

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Programs

Mathematica

A230311 Numbers n such that 1^(kn) + 2^(kn) + ... + (kn)^(kn) == k (mod kn) for some k; that is, numbers n such that A031971(kn) == k (mod k*n) for some k.

A229312 Numbers n such that A031971(47058n) == n (mod 47058n).

A231562 Numbers n such that A031971(8490421583559688410706771261086n) == n (mod 8490421583559688410706771261086n).

A230313 Numbers n such that A031971(47058n) <> n (mod 47058n).