A229303 -id:A229303 - cp's OEIS frontend

A229309 Primitive numbers in A229305.

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*#] == # &]

A229310 Primitive numbers in A229306.

Original entry on oeis.org

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

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

Original entry on oeis.org

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).

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*#] == # &]

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*#] == # &]

A262978 Exponents n such that 2^n-1 and 2^n+1 are squarefree.

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, 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

Offset: 1

Juri-Stepan Gerasimov, May 01 2016

			a(4) = 5 because 2^5 - 1 = 31 and 2^5 + 1 = 33 are squarefree numbers.

Amiram Eldar, Table of n, a(n) for n = 1..608

Cf. A000961, A229303, A269758.

[n: n in [1..120] | IsSquarefree(2^n-1) and IsSquarefree(2^n+1)];

Select[Range[120],AllTrue[2^#+{1,-1},SquareFreeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 20 2019 *)

is(n)=issquarefree(2^n-1) && issquarefree(2^n+1) \\ Charles R Greathouse IV, May 02 2016

2^a(n) = A269758(n).

cp's OEIS Frontend

A229309 Primitive numbers in A229305.

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Mathematica

A229310 Primitive numbers in A229306.

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Mathematica

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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Extensions

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

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Mathematica

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

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Mathematica

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

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Programs

Mathematica

A262978 Exponents n such that 2^n-1 and 2^n+1 are squarefree.

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Magma

Mathematica

PARI

Formula

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.

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

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).