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 1-8 of 8 results.

A114978 Numbers k such that (j^k + k^j) == 0 (mod k+j), j=3 case.

Original entry on oeis.org

1, 3, 5, 6, 7, 8, 9, 15, 21, 24, 27, 33, 36, 39, 51, 63, 69, 75, 81, 87, 99, 105, 111, 114, 118, 123, 135, 153, 165, 171, 183, 207, 213, 219, 231, 243, 249, 255, 267, 279, 309, 315, 348, 351, 363, 375, 387, 393, 399, 423, 435, 453, 465, 471, 495, 501
Offset: 1

Views

Author

Zak Seidov, Feb 22 2006

Keywords

Links

Crossrefs

Cf. A114977, A114979, A114980, A114981.

Programs

  • Mathematica
    Select[Range[600],Divisible[3^#+#^3,#+3]&] (* Harvey P. Dale, Nov 26 2019 *)
  • PARI
    isok(k, j=3) = (j^k+k^j) % (k+j) == 0; \\ Michel Marcus, Oct 10 2013

Extensions

More terms from Michel Marcus, Oct 10 2013

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

A156038 Odd numbers k with the property that (2^k + k^2) == 0 (mod (k+2)).

Original entry on oeis.org

1, 32377, 484177, 3094247, 10908137, 15612137, 16750127, 17363927, 24827519, 34030327, 184923407, 219087767, 240654017, 430450337, 814220815, 989880257, 1040713247, 1257956527, 1839451007, 2255176457, 2314599527, 3034111727, 3254592497, 3534277217, 3569817127
Offset: 1

Views

Author

Zak Seidov, Oct 30 2009

Keywords

Comments

Odd terms in A114977.

Links

Crossrefs

Cf. A114977.

Programs

A156048 Odd prime numbers p with the property that (2^p + p^2) == 0 (mod (p+2)).

Original entry on oeis.org

32377, 10908137, 34030327, 4860035567, 7656800897, 13398374537, 41162676047, 49277997407, 169906146727, 429221023247, 517971170207, 1587472376807, 1692526508927, 2687542461767, 2827976112047, 2918988711407, 3538852646177, 3843606175697, 3868058416007, 4090897510247
Offset: 1

Views

Author

Zak Seidov, Oct 31 2009

Keywords

Comments

Subsequence of A114977 and of A156038. - Michel Marcus, Oct 19 2013

Links

Crossrefs

Cf. A114977, A156038.

Programs

  • PARI
    isok(n) = isprime(n) && (n != 2) && ((2^n+n^2)% (n+2) == 0); \\ Michel Marcus, Oct 19 2013
  • PARI
    is(n)=Mod(2,n+2)^n==-4 && n>2 && isprime(n) \\ Charles R Greathouse IV, Oct 19 2013

Extensions

a(5) from Zak Seidov, Oct 31 2009
a(6)-a(8) from Juri-Stepan Gerasimov, Jan 16 2020
Terms a(9) and beyond from Giovanni Resta, Jan 23 2020

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 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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