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.

Showing 1-6 of 6 results.

A114977 Numbers k such that (j^k + k^j) == 0 (mod k+j), j=2 case.

Original entry on oeis.org

1, 2, 8, 128, 2144, 4808, 12872, 14168, 32377, 33672, 45992, 116192, 185768, 186824, 271208, 426008, 484177, 524288, 601352, 612768, 755792, 996032, 1878368, 2262752, 3094247, 4325960, 4810808, 6331808, 6707352, 10037792, 10908137, 11475128, 12672992, 13705232
Offset: 1

Views

Author

Zak Seidov, Feb 22 2006

Keywords

Comments

From Robert G. Wilson v, Aug 02 2021: (Start)
Prime terms: 2, then A156048.
The exponents of the powers of two which are terms: 0, 1, then A014741(n)+1.
The vast majority of terms are congruent to 8 (mod 24); no terms are congruent to 4 (mod 6) nor to 3 (mod 10).
(End)

Links

Crossrefs

Cf. A156038 (odd terms), A156048 (odd prime terms).
Cf. A014741, A114978, A114979, A114980, A114981.

Programs

Extensions

a(9)-a(22) from Michel Marcus, Oct 10 2013
a(23)-a(34) from Hiroaki Yamanouchi, Sep 26 2015

A114979 Numbers n such that (j^k + k^j) == 0 (mod k+j), j=4 case.

Original entry on oeis.org

1, 4, 12, 25, 28, 30, 60, 61, 109, 124, 252, 478, 529, 1024, 3273, 6172, 14881, 24700, 28732, 29701, 32509, 41437, 47569, 70009, 83209, 86629, 88177, 99712, 131161, 161473, 172092, 224389, 280869, 281509, 419017, 640129, 691840, 704089, 983641, 1048588
Offset: 1

Views

Author

Zak Seidov, Feb 22 2006

Keywords

Comments

a(755) > 10^11. - Hiroaki Yamanouchi, Sep 26 2015

Links

Crossrefs

Cf. A114977, A114978, A114980, A114981.

Programs

  • PARI
    isok(k, j=4) = (j^k+k^j) % (k+j) == 0; \\ Michel Marcus, Oct 10 2013

Extensions

a(11)-a(28) from Michel Marcus, Oct 10 2013
a(29)-a(40) from Hiroaki Yamanouchi, Sep 26 2015

A114980 Numbers k such that (j^k + k^j) == 0 (mod k+j), j=5 case.

Original entry on oeis.org

1, 3, 5, 7, 14, 15, 19, 20, 21, 23, 25, 26, 35, 45, 50, 55, 65, 73, 95, 105, 115, 119, 120, 125, 145, 165, 185, 195, 215, 225, 245, 270, 275, 285, 295, 305, 325, 350, 365, 385, 405, 425, 435, 437, 465, 495, 525, 527, 545, 585, 595, 605, 620, 621, 645
Offset: 1

Views

Author

Zak Seidov, Feb 22 2006

Keywords

Links

Crossrefs

Cf. A114977, A114978, A114979, A114981.

Programs

  • PARI
    isok(k, j=5) = (j^k+k^j) % (k+j) == 0; \\ Michel Marcus, Oct 10 2013
  • PARI
    isok(k, j=5) = Mod(j, k+j)^k+Mod(k, k+j)^j == 0; \\ Michel Marcus, Aug 07 2021

Extensions

More terms from Michel Marcus, Oct 10 2013

A114981 Numbers k such that (j^k + k^j) == 0 (mod k+j), j=6 case.

Original entry on oeis.org

1, 3, 6, 10, 12, 15, 18, 21, 26, 30, 42, 48, 57, 58, 62, 66, 68, 71, 72, 75, 87, 90, 102, 138, 156, 183, 186, 210, 216, 228, 237, 273, 282, 291, 318, 426, 476, 480, 561, 570, 576, 606, 642, 645, 660, 696, 701, 723, 831, 858, 951, 966, 1290, 1306, 1452
Offset: 1

Views

Author

Zak Seidov, Feb 22 2006

Keywords

Links

Crossrefs

Cf. A114977, A114978, A114979, A114980.

Programs

  • PARI
    isok(k, j=6) = (j^k+k^j) % (k+j) == 0; \\ Michel Marcus, Oct 10 2013
  • PARI
    isok(k, j=6) = Mod(j, k+j)^k+Mod(k, k+j)^j == 0; \\ Michel Marcus, Aug 07 2021

Extensions

More terms from Michel Marcus, Oct 10 2013

A242899 Least number k > 1 such that (n^k+k^n)/(k+n) is an integer.

Original entry on oeis.org

2, 2, 3, 4, 3, 3, 3, 2, 3, 6, 8, 4, 11, 5, 3, 16, 7, 6, 5, 5, 3, 10, 5, 3, 4, 5, 3, 4, 11, 4, 7, 11, 3, 30, 5, 3, 7, 19, 3, 10, 7, 6, 7, 11, 5, 12, 14, 6, 7, 5, 3, 12, 13, 9, 5, 8, 6, 6, 11, 4, 4, 6, 3, 64, 5, 6, 10, 6, 3, 10, 6, 6, 5, 37, 3, 30, 7, 12, 7, 20, 3, 40, 19, 9
Offset: 1

Views

Author

Derek Orr, May 25 2014

Keywords

Comments

a(n) <= n for all n > 1.

Examples

			(1^2+2^1)/(2+1) = 3/3 = 1 is an integer. Thus a(1) = 2.

Links

Crossrefs

Cf. A114977, A114978, A242872.

Programs

  • Mathematica
    lnk[n_]:=Module[{k=2},While[!IntegerQ[(n^k+k^n)/(k+n)],k++];k]; Array[lnk,90] (* Harvey P. Dale, Sep 02 2015 *)
  • PARI
    a(n)=if(n==1, 2, for(k=2, n, s=(n^k+k^n)/(k+n); if(floor(s)==s, return(k))))
n=1; while(n<100, print(a(n)); n+=1)
  • PARI
    a(n) = my(k=2); while (denominator((n^k+k^n)/(k+n))!=1, k++); k; \\ Michel Marcus, Jun 03 2021

A346744 The number of congruences k^(n-k) + (n-k)^k == 0 (mod n) with 0 < k < n.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 2, 5, 4, 7, 4, 7, 2, 3, 2, 9, 2, 7, 6, 7, 8, 3, 2, 15, 6, 9, 10, 11, 4, 17, 4, 17, 4, 9, 8, 15, 2, 5, 10, 15, 2, 13, 4, 11, 6, 3, 2, 19, 8, 15, 2, 11, 2, 19, 12, 19, 8, 11, 4, 19, 4, 5, 14, 33, 6, 13, 6, 17, 2, 25, 2, 31, 2, 9, 12, 11, 6, 29, 4
Offset: 1

Views

Author

Robert G. Wilson v, Jul 31 2021

Keywords

Comments

Of course for any n, k being equal to either 1 or n-1 would work.

Links

Crossrefs

Cf. A114977, A114978, A114979, A114980, A114981.

Programs

  • Maple
    a:= n-> add(`if`(k&^(n-k)+(n-k)&^k mod n=0, 1, 0), k=1..n-1):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 06 2021
  • Mathematica
    f[n_] := Block[{c = 0, k = 1}, While[k < n, If[ Mod[ PowerMod[k, n - k, n] + PowerMod[n - k, k, n], n] == 0, c++]; k++]; c]; Array[f@# &, 100]
  • PARI
    a(n) = sum(k=1, n-1, Mod(k, n)^(n-k) + Mod(n-k, n)^k == 0); \\ Michel Marcus, Aug 06 2021
  • Python
    def a(n): return sum((k**(n-k) + (n-k)**k)%n == 0 for k in range(1, n))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 31 2021
Showing 1-6 of 6 results.

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