A015951 -id:A015951 - cp's OEIS frontend

A119679 a(n) = least k such that the remainder when 5^k is divided by k is n.

2, 3, 22, 4769, 7, 15853, 114, 9, 28, 35, 14, 1328467, 68, 111, 1555, 9569200211, 76, 2030227, 49, 21, 299, 1097122717, 51, 546707, 26, 27, 121, 529, 596, 3095, 138, 93, 136, 34723, 45, 589, 198, 87, 18142961, 595, 292, 319, 318, 117, 55, 20485243, 91

Offset: 1

Ryan Propper, Jun 12 2006

nonn

From Alexander Adamchuk, Jan 31 2007: (Start)
a(n) > n.
For numbers n such that a(n-1) = n, see A015951 except first term. (End)
a(58) <= 16860204577843069 from Joe K. Crump (joecr(AT)carolina.rr.com), Feb 06 2007

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found [This file is now out of date - see the extension lines below. - _N. J. A. Sloane_, May 22 2010]

Cf. A015951, A036236, A078457, A119678, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821.

Do[k = 1; While[PowerMod[5, k, k] != n, k++ ]; Print[k], {n, 30}]
Table[0, {10000}]; k = 1; lst = {}; While[k < 5000000000, a = PowerMod[5, k, k]; If[ a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a,k}]]; k++ ]; t (* changed (to reflect the new limits) by Robert G. Wilson v, Jul 14 2009 *)

Revised by Max Alekseyev, Sep 25 2007
a(172) = 26598818717 = 23 * 593 * 1039 * 1877, a(288) = 9158745413 = 241 * 347 * 109519, a(518) = 33288260241 = 3 * 43 * 258048529, a(558) = 7722115807 = 7 * 157 * 7026493 from Daniel Morel, May 18 2010
a(848) = 6672480963 = 3 * 241 * 9228881 from Daniel Morel, May 26 2010
a(416) = 10545901269 from Daniel Morel, Jul 05 2010
a(948) = 146246024857 from Daniel Morel, Jul 12 2010
a(822) = 466661006683 from Daniel Morel, Aug 24 2010

A015949 Numbers k such that k | 3^k + 1.

Original entry on oeis.org

1, 2, 10, 50, 250, 1250, 5050, 6250, 11810, 25250, 31250, 59050, 126250, 156250, 295250, 510050, 631250, 750250, 781250, 1476250, 2125250, 2550250, 3156250, 3751250, 3906250, 5964050, 7381250, 10626250, 12751250, 13947610, 15781250

Offset: 1

Robert G. Wilson v

nonn

a(n) mod 20 = 10 for n >= 3. - G. C. Greubel, Nov 05 2018
This sequence is infinite, because for n > 1, 3^a(n) + 1 is in this sequence. - Jinyuan Wang, Nov 06 2018
For the provided data, if k is a term then p*k is a term where p is an odd divisor of k. - David A. Corneth, Nov 06 2018

Giovanni Resta, Table of n, a(n) for n = 1..180 (first 100 terms from G. C. Greubel)

Cf. A034472 (3^n+1).
Cf. A006521 (k | 2^k + 1), A015950 (k | 4^k + 1), A015951 (k | 5^k + 1).
Column k=3 of A333429.

[n: n in [1..2*10^7]| Modexp(3, n, n)+1 eq n]; // Vincenzo Librandi, Nov 01 2018

Do[If[PowerMod[3, n, n] + 1 == n, Print[n]], {n, 1, 10^7}] (* Jinyuan Wang, Nov 01 2018 *)
Select[Range[16*10^6],PowerMod[3,#,#]==#-1&] (* Harvey P. Dale, Dec 15 2024 *)

for(n=1, 10^7, if(Mod(3, n)^n==-1, print1(n, ", "))) \\ Jinyuan Wang, Nov 01 2018

Corrected by David W. Wilson

A015891 Numbers k such that k | 5^k + 5.

Original entry on oeis.org

1, 2, 5, 6, 10, 30, 70, 1565, 2806, 3126, 51670, 58290, 214405, 285286, 378258, 1854766, 2170486, 2222122, 2247610, 3463230, 4147522, 5942526, 9381126, 14818486, 15743890, 20162858, 34087054, 34838686, 38742166, 71067430

Offset: 1

Robert G. Wilson v

nonn

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A067946 = numbers n such that n divides 5^n-1. Cf. A015951 = numbers n such that n | 5^n + 1.

Cf. A006517, A015888, A015889, A015892, A015893, A015897, A015898, A015902, A015903, A015904, A015905, A015906.

Select[Range[1000000], IntegerQ[(PowerMod[5,#,# ]+5)/# ]&] (* Alexander Adamchuk *)

Corrected by Alexander Adamchuk, Nov 04 2006

A333429 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, 3, 0, 1, 2, 9, 0, 1, 5, 10, 27, 0, 1, 2, 25, 50, 81, 0, 1, 7, 3, 125, 250, 171, 0, 1, 2, 49, 9, 205, 1250, 243, 0, 1, 3, 10, 203, 21, 625, 5050, 513, 0, 1, 2, 9, 50, 343, 26, 1025, 6250, 729, 0, 1, 11, 5, 27, 250, 1379, 27, 2525, 11810, 1539, 0

Offset: 1

Alois P. Heinz, Mar 20 2020

			Square array A(n,k) begins:
  1,    1,     1,    1,   1,    1,     1,   1,    1,     1, ...
  2,    3,     2,    5,   2,    7,     2,   3,    2,    11, ...
  0,    9,    10,   25,   3,   49,    10,   9,    5,   121, ...
  0,   27,    50,  125,   9,  203,    50,  27,   25,   253, ...
  0,   81,   250,  205,  21,  343,   250,  57,   82,  1331, ...
  0,  171,  1250,  625,  26, 1379,  1250,  81,  125,  2783, ...
  0,  243,  5050, 1025,  27, 1421,  2810, 171,  625,  5819, ...
  0,  513,  6250, 2525,  63, 2401,  5050, 243, 2525, 11891, ...
  0,  729, 11810, 3125,  81, 5887,  6250, 513, 3125, 14641, ...
  0, 1539, 25250, 5125, 147, 9653, 14050, 729, 3362, 30613, ...

Alois P. Heinz, Antidiagonals n = 1..20, flattened

Columns k=1-16 give: A130779 (for n>=1), A006521, A015949, A015950, A015951, A015953, A015954, A015955, A015957, A015958, A015960, A015961, A015963, A015965, A015968, A015969.
Rows n=1-2 give: A000012, A092067.
Main diagonal gives A333430.
Cf. A333432.

A:= proc() local h, p; p:= proc() [1] end;
      proc(n, k) if k=1 then `if`(n<3, n, 0) else
        while nops(p(k)) 0 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);

dmax = 12;
mmax = 2^(dmax+3);
col[k_] := col[k] = Select[Range[mmax], Divisible[k^#+1, #]&];
A[n_, k_] := If[n>2 && k==1, 0, col[k][[n]]];
Table[A[n, d-n+1], {d, 1, dmax}, {n, 1, d}] // Flatten (* Jean-François Alcover, Jan 05 2021 *)

A123047 Numbers k that divide 5^k + 4.

Original entry on oeis.org

1, 3, 129, 60767, 76433163, 454034821, 26675718567, 164304369911289

Offset: 1

Alexander Adamchuk, Nov 04 2006

k must be odd since any power of 5 plus 4 is odd. - Robert G. Wilson v, Nov 14 2006
a(9) > 10^15. - Max Alekseyev, Oct 17 2016
Large term (may not be the next one): 3014733401203184049549. - Max Alekseyev, Oct 18 2013

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), A125949 (k=4), this sequence (k=-4), A123091 (k=5), A015891 (k=-5), A277350 (k=6), A277348 (k=-6).

Do[ If[ PowerMod[5, 2n - 1, 2n - 1] + 5 == 2n, Print[2n - 1]], {n, 10^9}] (* Robert G. Wilson v *)

is(n)=Mod(5,n)^n==-4 \\ Charles R Greathouse IV, Nov 04 2016

a(4) and a(5) from Robert G. Wilson v, Nov 14 2006
a(7) from Ryan Propper, Mar 23 2007
a(8) from Max Alekseyev, Oct 17 2016

A015954 Numbers k such that k | 7^k + 1.

Original entry on oeis.org

1, 2, 10, 50, 250, 1250, 2810, 5050, 6250, 14050, 25250, 31250, 40210, 70250, 126250, 156250, 201050, 351250, 510050, 631250, 650050, 781250, 789610, 1005250, 1265050, 1419050, 1756250, 2550250, 3156250, 3250250, 3906250, 3948050, 5026250, 6325250, 7095250, 8781250, 9478130

Offset: 1

Robert G. Wilson v

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..100

Solutions to b^k == -1 (mod k): A006521 (b=2), A015949 (b=3), A015950 (b=4), A015951 (b=5), A015953 (b=6), this sequence (b=7), A015955 (b=8), A015957 (b=9), A015958 (b=10), A015960 (b=11), A015961 (b=12), A015963 (b=13), A015965 (b=14), A015968 (b=15), A015969 (b=16).
Cf. A277370, A277401.
Column k=7 of A333429.

A015960 Numbers k such that k | 11^k + 1.

Original entry on oeis.org

1, 2, 3, 9, 27, 81, 111, 122, 243, 333, 729, 999, 2187, 2997, 4107, 6561, 7442, 8991, 10233, 12321, 13203, 19683, 24753, 26973, 30699, 36963, 39609, 59049, 74259, 80919, 89426, 92097, 110889, 118341, 118827, 151959, 177147, 222777

Offset: 1

Robert G. Wilson v

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..200

Solutions to b^k == -1 (mod k): A006521 (b=2), A015949 (b=3), A015950 (b=4), A015951 (b=5), A015953 (b=6), A015954 (b=7), A015955 (b=8), A015957 (b=9), A015958 (b=10), this sequence (b=11), A015961 (b=12), A015963 (b=13), A015965 (b=14), A015968 (b=15), A015969 (b=16).
Cf. A333134.
Column k=11 of A333429.

Select[Range[250000],PowerMod[11,#,#]==#-1&] (* Harvey P. Dale, Nov 09 2022 *)

A123062 Numbers k that divide 5^k + 2.

Original entry on oeis.org

1, 7, 51373, 78127, 138943, 620299, 2842933, 137422693, 2259290321, 413879131637, 434757575329, 915535274009, 14864856896743

Offset: 1

Alexander Adamchuk, Nov 04 2006

No further terms up to 10^15. Larger term: 64629734103979763971. - Max Alekseyev, Oct 15 2016

Solutions to 5^n == k (mod n): A067946 (k=1), A015951 (k=-1), A124246 (k=2), this sequence (k=-2), A123061 (k=3), A123052 (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,#,# ]+2)/# ]&]

is(n)=Mod(5,n)^n==-2 \\ Charles R Greathouse IV, Nov 04 2016

More terms from Farideh Firoozbakht, Nov 18 2006
a(9) from Ryan Propper, Jan 29 2007
a(10)-a(13) from Max Alekseyev, Jul 28 2009, Oct 15 2016

A123091 Numbers k such that k divides 5^k - 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 11, 13, 15, 17, 19, 20, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 124, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 190, 191, 193, 197, 199, 211, 217, 223, 227, 229, 233, 239

Offset: 1

Alexander Adamchuk, Nov 04 2006

nonn

All primes are the terms of a(n). Nonprimes in a(n) are listed in A122782(n) = {1,4,10,15,20,65,124,190,217,310,435,561,781,...}. All pseudoprimes to base 5 are the terms of a(n). They are listed in A005936(n) = {4,124,217,561,781,...}. Numbers n up to 10^6 such that n divides 5^n + 5 are {1,2,5,6,10,30,70,1565,2806,3126,51670,58290,214405,285286,378258}.

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

Cf. A122782 (nonprimes n such that 5^n==5 (mod n)).
Cf. A005936 (pseudoprimes to base 5).
Cf. A067946 (numbers n such that n divides 5^n-1).
Cf. A015951 (numbers n such that n | 5^n + 1).

Select[Range[1000], IntegerQ[(PowerMod[5,#,# ]-5)/# ]&]

is(n)=Mod(5,n)^n==5 \\ Charles R Greathouse IV, Nov 04 2016

A124246 Numbers k that divide 5^k - 2.

Original entry on oeis.org

1, 3, 123, 202884639, 242405133, 92273577267, 2670733723929, 81035221987959

Offset: 1

Farideh Firoozbakht, Nov 19 2006

nonn

No other terms below 10^15. Some larger terms: 60092749466423900486673922957841, 401021769827858799355246286337987697472836927856337282726789534497163. - Max Alekseyev, Oct 15 2016

Solutions to 5^n == k (mod n): A067946 (k=1), A015951 (k=-1), this sequence (k=2), A123062 (k=-2), A123061 (k=3), A123052 (k=-3), A125949 (k=4), A123047 (k=-4), A123091 (k=5), A015891 (k=-5), A277350 (k=6), A277348 (k=-6).

Do[If[Mod[(PowerMod[5,n,n]-2),n]==0,Print[n]],{n,1000000000}]

is(n)=Mod(5,n)^n==2 \\ Charles R Greathouse IV, Nov 04 2016

a(6)-a(8) from Max Alekseyev, Jul 28 2009, Jun 02 2010, Oct 15 2016

cp's OEIS Frontend

A119679 a(n) = least k such that the remainder when 5^k is divided by k is n.

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A015949 Numbers k such that k | 3^k + 1.

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A015891 Numbers k such that k | 5^k + 5.

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A333429 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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A123047 Numbers k that divide 5^k + 4.

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A015954 Numbers k such that k | 7^k + 1.

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A015960 Numbers k such that k | 11^k + 1.

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A123062 Numbers k that divide 5^k + 2.

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A123091 Numbers k such that k divides 5^k - 5.

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