A067946 -id:A067946 - cp's OEIS frontend

A014957 Positive integers k that divide 16^k - 1.

1, 3, 5, 9, 15, 21, 25, 27, 39, 45, 55, 63, 75, 81, 105, 117, 125, 135, 147, 155, 165, 171, 189, 195, 205, 225, 243, 273, 275, 315, 333, 351, 375, 405, 441, 465, 495, 507, 513, 525, 567, 585, 605, 609, 615, 625, 657, 675, 729, 735, 775, 819, 825, 855, 903

Offset: 1

Olivier Gérard

nonn

Also, positive integers k that divide A014931(k).

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A014956, A177805, A177807, A128358, A128360

Cf. A128396, A177916

Join[{1},Select[Range[1000],PowerMod[16,#,#]==1&]] (* Harvey P. Dale, Jun 12 2024 *)

A014957_list = [n for n in range(1,10**6) if n == 1 or pow(16,n,n) == 1] # Chai Wah Wu, Mar 25 2021

Edited by Max Alekseyev, Sep 10 2011

A123052 Numbers k that divide 5^k + 3.

Original entry on oeis.org

1, 2, 4, 14, 628, 11524, 16814, 188404, 441484, 2541014, 3984724, 172315684, 208268941, 40874725514, 280454588548, 489850370956, 1235856817732, 62479203805793, 95467808763364, 116016015619396, 396249210287836

Offset: 1

Alexander Adamchuk, Nov 04 2006

No other terms below 10^15. A larger term: 783847656467936404. - Max Alekseyev, Oct 16 2016

Solutions to 5^n == k (mod n): A067946 (k=1), A015951 (k=-1), A124246 (k=2), A123062 (k=-2), A123061 (k=3), this sequence (k=-3), A125949 (k=4), A123047 (k=-4), A123091 (k=5), A015891 (k=-5), A277350 (k=6), A277348 (k=-6).

Select[Range[1000000], IntegerQ[(PowerMod[5,#,# ]+3)/# ]&]

is(n)=Mod(5,n)^n==-3 \\ Charles R Greathouse IV, Apr 06 2014

a(10)-a(13) from Ryan Propper, Dec 30 2006, Jan 02 2007
More terms from Lars Blomberg, Nov 25 2011
Terms a(14) onwards were reported incorrect by Toshitaka Suzuki, and  have been deleted. - N. J. A. Sloane, Mar 18 2014
a(14)-a(17) from Toshitaka Suzuki, Mar 18 2014, Apr 03 2014
a(18)-a(21) from Max Alekseyev, Oct 16 2016

A125949 Numbers k that divide 5^k - 4.

Original entry on oeis.org

1, 4769, 8563651, 300414792131, 2353957351049, 15960089894129, 452045914836301, 657236915690111

Offset: 1

Alexander Adamchuk, Feb 04 2007

No other terms below 10^15. - Max Alekseyev, Oct 17 2016

Solutions to 5^n == k (mod n): A067946 (k=1), A015951 (k=-1), A124246 (k=2), A123062 (k=-2), A123061 (k=3), A123052 (k=-3), this sequence (k=4), A123047 (k=-4), A123091 (k=5), A015891 (k=-5), A277350 (k=6), A277348 (k=-6).

a(1) = 1; Do[ If[ PowerMod[5, 2n - 1, 2n - 1] - 4 == 0, Print[2n - 1]], {n,10^9}]

is(n)=Mod(5,n)^n==4 \\ Charles R Greathouse IV, May 15 2013

a(4)-a(8) from Max Alekseyev, Jun 09 2010, Oct 17 2016

A277350 Positive integers n such that 5^n == 6 (mod n).

Original entry on oeis.org

1, 15853, 5520343, 111966563, 2232207889, 5551501871

Offset: 1

Seiichi Manyama, Oct 10 2016

No other terms below 10^15. - Max Alekseyev, Oct 18 2016

Cf. Solutions to 5^n == k (mod n): A277348 (k=-6), A015891 (k=-5), A123047 (k=-4), A123052 (k=-3), A123062 (k=-2), A015951 (k=-1), A067946 (k=1), A124246 (k=2), A123061 (k=3), A125949 (k=4), A123091 (k=5), this sequence (k=6).

isok(n) = Mod(5, n)^n == 6; \\ Michel Marcus, Oct 10 2016

A277348 Positive integers n such that n | (5^n + 6).

Original entry on oeis.org

1, 11, 341, 581337017, 7202608727, 27146455379, 1358496201131, 9843739213499, 172392038905691

Offset: 1

Seiichi Manyama, Oct 10 2016

No other terms below 10^15. - Max Alekseyev, Oct 17 2016

			5^11 + 6 = 48828131 = 11 * 4438921, so 11 is a term.

Cf. A066603.

Cf. Solutions to 5^n == k (mod n): this sequence (k=-6), A015891 (k=-5), A123047 (k=-4), A123052 (k=-3), A123062 (k=-2), A015951 (k=-1), A067946 (k=1), A124246 (k=2), A123061 (k=3), A125949 (k=4), A123091 (k=5), A277350 (k=6).

isok(n) = Mod(5, n)^n == -6; \\ Michel Marcus, Oct 10 2016

A066603(a(n)) = a(n) - 6 for n > 1.

a(5)-a(9) from Max Alekseyev, Oct 17 2016

A083528 a(n) = 5^n mod 2*n.

Original entry on oeis.org

1, 1, 5, 1, 5, 1, 5, 1, 17, 5, 5, 1, 5, 25, 5, 1, 5, 1, 5, 25, 41, 25, 5, 1, 25, 25, 53, 9, 5, 25, 5, 1, 59, 25, 45, 1, 5, 25, 47, 65, 5, 1, 5, 9, 35, 25, 5, 1, 19, 25, 23, 1, 5, 1, 45, 81, 11, 25, 5, 25, 5, 25, 125, 1, 5, 49, 5, 81, 125, 65, 5, 1, 5, 25, 125, 17, 3, 25, 5, 65, 161, 25, 5, 1, 65

Offset: 1

Labos Elemer, Apr 30 2003

a(n) = 1 iff n is in A067946. - Robert Israel, Dec 26 2014

			a(3) = 5 because 5^3 = 125 and 125 == 5 mod (2 * 3).
a(4) = 1 because 5^4 = 625 and 625 == 1 mod (2 * 4).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000079, A000244, A000351, A000420, A082511.

Cf. A067946, A083529, A083530.

[Modexp(5, n, 2*n): n in [1..80]]; // Vincenzo Librandi, Oct 19 2018

seq(5 &^n mod (2*n), n = 1 .. 100); # Robert Israel, Dec 26 2014

Mathematica
```
Table[PowerMod[5, w, 2w], {w, 1, 100}]
```

vector(100, n, lift(Mod(5, 2*n)^n)) \\ Michel Marcus, Dec 29 2014

A083529 a(n) = 5^n mod 3*n.

Original entry on oeis.org

2, 1, 8, 1, 5, 1, 5, 1, 26, 25, 5, 1, 5, 25, 35, 1, 5, 1, 5, 25, 62, 25, 5, 1, 50, 25, 80, 37, 5, 55, 5, 1, 26, 25, 80, 1, 5, 25, 8, 25, 5, 1, 5, 97, 80, 25, 5, 1, 68, 25, 125, 1, 5, 1, 155, 25, 125, 25, 5, 145, 5, 25, 188, 1, 5, 181, 5, 13, 125, 205, 5, 1, 5, 25, 125, 169, 80, 181, 5

Offset: 1

Labos Elemer, Apr 30 2003

From Robert Israel, Dec 25 2014: (Start)
a(n) == (-1)^n mod 3.
a(n) = 1 if and only if n is even and in A067946.
For n > 3, a(n) = 5 if and only if n is odd and in A123091. (End)

			a(3) = 8 because 5^3 = 125 and 125 mod (3 * 3) = 8.
a(4) = 1 because 5^4 = 625 and 625 mod (3 * 4) = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000079, A000244, A000351, A000420, A082511, A083528, A083530, A067946, A123091.

[Modexp(5, n, 3*n): n in [1..80]]; // Vincenzo Librandi, Oct 19 2018

seq(5 &^n mod 3*n, n = 1 .. 1000); # Robert Israel, Dec 25 2014

Table[Mod[5^w, 3 * w], {w, 100}]
Table[PowerMod[5, n, 3n], {n, 80}] (* Harvey P. Dale, Jul 26 2014 *)

a(n)=lift(Mod(5,3*n)^n) \\ Charles R Greathouse IV, Oct 03 2016

A333432 A(n,k) is the n-th number m that divides k^m - 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 3, 4, 0, 5, 1, 2, 9, 8, 0, 6, 1, 5, 4, 21, 16, 0, 7, 1, 2, 25, 6, 27, 20, 0, 8, 1, 7, 3, 125, 8, 63, 32, 0, 9, 1, 2, 49, 4, 625, 12, 81, 40, 0, 10, 1, 3, 4, 343, 6, 1555, 16, 147, 64, 0, 11, 1, 2, 9, 8, 889, 8, 3125, 18, 171, 80, 0, 12

Offset: 1

Seiichi Manyama, Mar 21 2020

			Square array A(n,k) begins:
  1, 1,  1,   1,  1,     1,  1,     1,  1, ...
  2, 0,  2,   3,  2,     5,  2,     7,  2, ...
  3, 0,  4,   9,  4,    25,  3,    49,  4, ...
  4, 0,  8,  21,  6,   125,  4,   343,  8, ...
  5, 0, 16,  27,  8,   625,  6,   889, 10, ...
  6, 0, 20,  63, 12,  1555,  8,  2359, 16, ...
  7, 0, 32,  81, 16,  3125,  9,  2401, 20, ...
  8, 0, 40, 147, 18,  7775, 12,  6223, 32, ...
  9, 0, 64, 171, 24, 15625, 16, 16513, 40, ...

Seiichi Manyama, Antidiagonals n = 1..25, flattened
OEIS Wiki, 2^n mod n

Columns k=1-20 give: A000027, A063524, A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A177805, A014957, A177807, A128358, A333506, A128360.
Main diagonal gives A333433.
Cf. A333429, A333500.

A:= proc() local h, p; p:= proc() [1] end;
      proc(n, k) if k=2 then `if`(n=1, 1, 0) else
        while nops(p(k)) 1 do od;
          p(k):= [p(k)[], h]
        od; p(k)[n] fi
      end
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Mar 24 2020

A[n_, k_] := Module[{h, p}, p[_] = {1}; If[k == 2, If[n == 1, 1, 0], While[ Length[p[k]] < n, For[h = 1 + p[k][[-1]], Mod[k^h, h] != 1, h++]; p[k] = Append[p[k], h]]; p[k][[n]]]];
Table[A[n, 1+d-n], {d, 1, 12}, {n, 1, d}] // Flatten (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

A014959 Integers k such that k divides 22^k - 1.

Original entry on oeis.org

1, 3, 7, 9, 21, 27, 39, 49, 63, 81, 117, 147, 189, 243, 273, 343, 351, 441, 507, 567, 729, 819, 1029, 1053, 1143, 1323, 1521, 1701, 1911, 2187, 2401, 2457, 2943, 3081, 3087, 3159, 3429, 3549, 3969, 4401, 4563, 5103, 5733, 6561, 6591, 7203, 7371

Offset: 1

Olivier Gérard

nonn

Also, numbers n such that n divides s(n), where s(1)=1, s(k)=s(k-1)+k*22^(k-1) (cf. A014940).

Max Alekseyev, Table of n, a(n) for n = 1..69065

Integers n such that n divides b^n - 1: A067945 (b=3), A014945 (b=4), A067946 (b=5), A014946 (b=6), A067947 (b=7), A014949 (b=8), A068382 (b=9), A014950 (b=10), A068383 (b=11), A014951 (b=12), A116621 (b=13), A014956 (b=14), A177805 (b=15), A014957 (b=16), A177807 (b=17), A128358 (b=18), A125000 (b=19), A128360 (b=20), A014960 (b=24).

nxt[{n_,s_}]:={n+1,s+(n+1)*22^n}; Transpose[Select[NestList[nxt,{1,1},7500], Divisible[ Last[#],First[#]]&]][[1]] (* Harvey P. Dale, Jan 27 2015 *)

Edited by Max Alekseyev, Nov 16 2019

A115976 Numbers k that divide 2^(k-2) + 1.

Original entry on oeis.org

1, 3, 49737, 717027, 9723611, 21335267, 32390921, 38999627, 43091897, 86071337, 101848553, 102361457, 228911411, 302948067, 370219467, 393664027, 455781089, 483464027, 1040406177, 1272206987, 2371678553, 2571052241, 2648052857, 3054713937, 3597613307, 3782971499, 3917903851, 4005163577, 5419912241

Offset: 1

Max Alekseyev, Mar 15 2006

nonn

Some larger terms: 4465786944074559659, 1440261542571735083956640176981881665928575750093930787551969

Max Alekseyev, Table of n, a(n) for n = 1..708 (contains all terms below 10^15)

The odd elements of A244673.

Cf. A006521, A006517, A069927, A067945, A067946, A067947, A068382, A068383, A014945, A014946, A014949, A092028.

lst = {}; Do[ If[ PowerMod[2, 2n - 3, 2n - 1] == 2n - 2, AppendTo[lst, 2n - 1]], {n, 10^9}]; lst (* Robert G. Wilson v, Apr 04 2006 *)

More terms from Robert G. Wilson v, Apr 04 2006
Terms a(24) onward from Max Alekseyev, Feb 03 2015
b-file corrected and extended by Max Alekseyev, Oct 27 2018

cp's OEIS Frontend

A014957 Positive integers k that divide 16^k - 1.

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A123052 Numbers k that divide 5^k + 3.

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A125949 Numbers k that divide 5^k - 4.

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A277350 Positive integers n such that 5^n == 6 (mod n).

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A277348 Positive integers n such that n | (5^n + 6).

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A083528 a(n) = 5^n mod 2*n.

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A083529 a(n) = 5^n mod 3*n.

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A333432 A(n,k) is the n-th number m that divides k^m - 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

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