cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

A327840 Numbers m that divide 4^m + 3.

Original entry on oeis.org

1, 7, 16387, 4509253, 24265177, 42673920001, 103949349763, 12939780075073
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 27 2019

Keywords

Comments

Number of solutions < 10^9 to k^n == k-1 (mod n): 1 (if k = 1), 188 (if k = 2, see A006521), 5 (if k = 3, see A015973), 5 (if k = 4, see this sequence), 5 (if k = 5), 10 (if k = 6), 10 (if k = 7), 7 (if k = 8), 5 (if k = 9), 8 (if k = 10), 11 (if k = 11), 8 (if k = 12), 9 (if k = 13), 4 (if k = 14), 3 (if k = 15), 6 (if k = 16), 7 (if k = 17), 7 (if k = 18), ...
a(9) > 10^15. - Max Alekseyev, Nov 10 2022

Crossrefs

Solutions to k^n == 1-k (mod n): A006521 (k = 2), A015973 (k = 3), this sequence (k = 4), A123047 (k = 5), A327943 (k = 6).
Solutions to 4^n == k (mod n): A000079 (k = 0), A015950 (k = -1), A014945 (k = 1), A130421 (k = 2), this sequence (k = -3), A130422 (k = 3).
Cf. A015940, A253208.

Programs

  • Magma
    [1] cat [n: n in [1..10^8] | Modexp(4,n,n) + 3 eq n];
  • Mathematica
    Select[Range[10^7], IntegerQ[(PowerMod[4, #, # ]+3)/# ]&] (* Metin Sariyar, Sep 28 2019 *)
  • PARI
    is(n)=Mod(4,n)^n==-3 \\ Charles R Greathouse IV, Sep 29 2019

Extensions

a(6)-a(7) from Giovanni Resta, Sep 29 2019
a(8) from Max Alekseyev, Nov 10 2022

A328033 Numbers m that divide 7^m + 6.

Original entry on oeis.org

1, 13, 793, 1943, 150341, 183793, 2348789, 26052527, 27982637, 54789869, 1588344433, 3928538029, 8115802931, 16936276919, 17786709541, 47778790033, 973094452518029
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 02 2019

Keywords

Comments

Conjecture: For k > 1, k^m == 1 - k (mod m) has infinite number of positive solutions.
Also includes 2073273696480171732497. - Giovanni Resta, Oct 04 2019

Crossrefs

Solutions to k^m == 1-k (mod m): A006521 (k = 2), A015973 (k = 3), A327840 (k = 4), A123047 (k = 5), A327943 (k = 6), this sequence (k = 7), A327468 (k = 8).
Cf. A253210 (7^n + 6).

Programs

  • Magma
    [1] cat [n: n in [1..10^8] | Modexp(7, n, n) + 6 eq n];

Extensions

a(12)-a(16) from Giovanni Resta, Oct 04 2019
a(17) from Max Alekseyev, Feb 07 2024

A327468 Numbers m that divide 8^m + 7.

Original entry on oeis.org

1, 3, 5, 25, 519, 290502305, 821808425, 979288025, 982989263, 25783323897, 27771237541, 31045665345, 65130752425, 3708883906025, 15079242289703, 973336048301405
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 04 2019

Keywords

Comments

Conjecture: For k > 1, k^m == 1-k (mod m) has an infinite number of positive solutions.
Integer m not divisible by 3 is a term if and only if 3m is a term of A240941. - Max Alekseyev, Feb 07 2024
Also terms 930486448009391617725 and 21036656390681764555645540794214294457925. - Giovanni Resta, Oct 04 2019
Other terms 71245661271703622047, 7093208961478946798805, 7807963392818324067361574236385. - Max Alekseyev, Feb 07 2024

Crossrefs

Solutions to k^m == 1-k (mod m): 1 (k = 1), A006521 (k = 2), A015973 (k = 3), A327840 (k = 4), A123047 (k = 5), A327943 (k = 6), A328033 (k = 7), this sequence (k = 8).
Cf. A240941, A253211.

Programs

  • Magma
    [m: m in [1..7] | (8^m + 7) mod m eq 0] cat [m: m in [8..10^8] | Modexp(8, m, m) + 7 eq m]; // Jon E. Schoenfield, Oct 05 2019
  • PARI
    isok(n) = Mod(8, n)^n==-7; \\ Michel Marcus, Oct 05 2019

Extensions

a(10)-a(13) from Giovanni Resta, Oct 04 2019
a(14)-a(16) from Max Alekseyev, Feb 07 2024

A292392 Numbers n such that n^2 divides (17^n + 1).

Original entry on oeis.org

1, 3, 9, 21, 39, 63, 117, 273, 819, 2067, 3081, 6201, 9243, 12807, 14469, 21567, 43407, 48711, 50877, 64701, 89649, 146133, 149331, 163293, 166491, 221169, 340977, 356139, 447993, 489879, 546819, 661401, 663507, 1022931, 1143051, 1165437, 1548183, 1639911, 1640457
Offset: 1

Views

Author

K. D. Bajpai, Sep 15 2017

Keywords

Comments

After a(1), all the terms are multiples of 3.
From Robert Israel, Sep 18 2017: (Start)
All terms are odd.
If m and n are terms then lcm(m,n) is a term.
If n is a term not divisible by 9, then 3n is a term. (End)

Examples

			3 appears is a term because 3^2 divides (17^3 + 1): 4914/9 = 546.
9 appears is a term because 9^2 divides (17^9 + 1): 118587876498/81 = 1464047858.

Crossrefs

Cf. A015951, A123047, A123052, A177813, A292331.

Programs

  • Maple
    A292392:= proc(n) if(17 &^ n+1)mod (n^2)=0  then RETURN (n); fi; end: seq(A292392(n), n=1..50000);
  • Mathematica
    Select[Range[50000], IntegerQ[(PowerMod[17, #, #^2] + 1)/#^2] &]
  • PARI
    for(n=1, 5e6, if (Mod(17, n^2)^n==-1, print1(n, ", ")));
  • PARI
    is(n) = Mod(17, n^2)^n==-1 \\ Felix Fröhlich, Sep 16 2017

A327763 Numbers k such that 4k - 1 divides 2^k + k^2.

Original entry on oeis.org

1, 5, 2071, 33421, 58061, 1977071, 7011161, 10144571, 10189721, 166311841, 173618771, 215766701, 240623221, 341359721, 341455271, 389192441, 490814481, 524456461, 585387477, 888576461, 1123952441, 1276706071, 2005030751, 2244832943, 2863311531, 3104590331, 3904529471
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 24 2019

Keywords

Comments

Primes: 5, 58061, 341455271,...

Crossrefs

Cf. A001580, A327636, A123047, A327943.

Programs

  • Mathematica
    Select[Range[10^5], Mod[PowerMod[2, #, 4*#-1] + #^2, 4*#-1] == 0 &] (* Giovanni Resta, Oct 09 2019 *)

Extensions

a(10)-a(27) from Giovanni Resta, Oct 09 2019

A328138 Numbers m that divide 9^m + 8.

Original entry on oeis.org

1, 17, 803, 1241, 20264753, 28214180783393, 228454543831049
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 04 2019

Keywords

Comments

Conjecture: For n > 1, k^n == 1-k (mod n) has an infinite number of positive solutions.
No term can be a multiple of 2, 3, 5, 7, or 13. Also 4879573990210017348077958628152400091281634488825721395187 is a term. - Giovanni Resta, Oct 07 2019
Also 6788776064693081883870036833 is a term. - Max Alekseyev, Dec 27 2024

Crossrefs

Subsequence of A008364.
Solutions to k^m == k-1 (mod m): 1 (k = 1), A006521 (k = 2), A015973 (k = 3), A327840 (k = 4), A123047 (k = 5), A327943 (k = 6), A328033 (k = 7), A327468 (k = 8), this sequence (k = 9).
Cf. A253212 (9^n + 8).

Programs

  • Magma
    [1] cat [n: n in [1..10^8] | Modexp(9, n, n) + 8 eq n];
  • PARI
    isok(n) = Mod(9, n)^n==-8; \\ Michel Marcus, Oct 05 2019

Formula

a(n) > 15n for large enough n. (Surely the sequence grows superlinearly, but I can't prove it.) - Charles R Greathouse IV, Dec 27 2024

Extensions

a(7) from Giovanni Resta confirmed and a(6) added by Max Alekseyev, Dec 27 2024
Previous Showing 11-16 of 16 results.

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