A067945 -id:A067945 - cp's OEIS frontend

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.

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

A242865 Numbers n such that 3^(n - 3) is congruent to 1 modulo n.

Original entry on oeis.org

3, 9299, 31903, 50963, 87043, 115918, 116891, 219827, 241043, 394243, 550243, 617503, 760243, 806623, 1029253, 1050787, 1458083, 1642798, 1899458, 2864755, 3205387, 3588115, 3839363, 4164578, 5041223, 5610583, 5834755, 5977555, 7837903, 8005558, 8067433, 8128823, 9007603, 9298903, 9449113, 9617443, 9835843

Offset: 1

Felix Fröhlich, May 24 2014

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..372 (terms 1..163 from Felix Fröhlich)

Cf. A067945, A173572.

Intersection with A033553 gives A277344.

Select[Range[10^4], Mod[3^(# - 3), #] == 1 &] (* Alonso del Arte, May 27 2014 *)

for(n=3, 10^6, if(Mod(3, n)^(n-3)==1, print1(n, ", ")))

A277628 Positive integers n such that 3^n == 6 (mod n).

Original entry on oeis.org

1, 3, 21, 936340943, 10460353197, 9374251222371, 23326283250291, 615790788171551

Offset: 1

Dmitry Ezhov, Oct 24 2016

No other terms below 10^15. - Max Alekseyev, Sep 12 2017

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), this sequence (k=6), A277126 (k=7), A277630 (k=8), A277274 (k=11).

PARI
```
isok(n) = Mod(3, n)^n == Mod(6, n);
```

a(6)-a(8) from Max Alekseyev, Sep 12 2017

A277630 Positive integers n such that 3^n == 8 (mod n).

Original entry on oeis.org

1, 5, 2352527, 193841707, 17126009179703, 380211619942943

Offset: 1

Dmitry Ezhov, Oct 24 2016

No other terms below 10^15. - Max Alekseyev, Sep 13 2017

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), A277628 (k=6), A277126 (k=7), this sequence (k=8), A277274 (k=11).

PARI
```
isok(n) = Mod(3, n)^n == Mod(8, n);
```

a(5)-a(6) established by Max Alekseyev, Sep 13 2017

A093547 Numbers k such that k divides 3^(k^2) - 1.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 20, 32, 40, 50, 64, 68, 80, 100, 110, 128, 136, 160, 164, 200, 220, 250, 256, 272, 320, 328, 340, 400, 440, 500, 512, 544, 550, 610, 640, 656, 680, 772, 800, 820, 880, 1000, 1010, 1024, 1088, 1100, 1156, 1210, 1220, 1250, 1280, 1312, 1360

Offset: 1

Farideh Firoozbakht, Mar 31 2004

nonn

This sequence is closed under multiplication, i.e., if x and y are terms then so is x*y.

A067945 is a subsequence of this sequence. A067945 is also closed under multiplication. In fact if m is an integer and k is a natural number then the sequence " n divides m^(n^k) - 1 " is a subsequence of the sequence " n divides m^n^(k+1)- 1 " and both are closed under multiplication.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A067945, A093546, A006521.

v={};Do[If[IntegerQ[(3^n^2-1)/n], v=Append[v, n];Print[v]], {n, 2500}]
Select[Range[1400],Divisible[3^#^2-1,#]&] (* Harvey P. Dale, Nov 04 2015 *)

isok(k) = Mod(3, k)^(k^2) == 1; \\ Amiram Eldar, May 26 2024

cp's OEIS Frontend

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

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Mathematica

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

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Mathematica

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A242865 Numbers n such that 3^(n - 3) is congruent to 1 modulo n.

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Mathematica

PARI

A277628 Positive integers n such that 3^n == 6 (mod n).

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PARI

Extensions

A277630 Positive integers n such that 3^n == 8 (mod n).

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PARI

Extensions

A093547 Numbers k such that k divides 3^(k^2) - 1.

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Mathematica

PARI