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 1-4 of 4 results.

A252041 Numbers m such that m - 3 divides m^m + 3.

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 85, 105, 136, 186, 262, 820, 1161, 2626, 2926, 4924, 10396, 11656, 19689, 27637, 33736, 36046, 42886, 42901, 53866, 55189, 82741, 95266, 103762, 106822, 127401, 135460, 251506, 366796, 375220, 413326, 466966, 531445, 553456, 568876
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 12 2014

Keywords

Comments

Numbers m such that (m^m + 3)/(m - 3) is an integer.
Most but not all terms are congruent to 4 modulo 6. - Robert G. Wilson v, Dec 19 2014
Note that m^m == 3^m (mod m-3). - Robert Israel, Dec 19 2014

Examples

			2 is in this sequence because (2^2 + 3)/(2 - 3) = -7 is an integer.
4 is in this sequence because (4^4 + 3)/(4 - 3) = 259 is an integer.
7 is not in the sequence because (7^7 + 3)/4 = 411773/2, which is not an integer.

Links

Crossrefs

Cf. ...............Numbers n such that x divides y, where:
...x......y....k = 0.....k = 1.....k = 2......k = 3.......
..n-k..n^n-k..A000027...A087156...A242787....A242788......
..n-k..n^n+k..A000027..see below..A249751..this sequence..
..n+k..n^n-k..A000027...A004275...A251603....A251862......
..n+k..n^n+k..A000027...A004273...A213382....A242800......
(For x=n-1 and y=n^n+1, the only terms are 0, 2 and 3. - David L. Harden, Dec 28 2014)

Programs

  • Magma
    [n: n in [4..50000] | Denominator((n^n+3)/(n-3)) eq 1];
  • Maple
    select(t -> 3 &^t + 3 mod (t-3) = 0, [1,2,$4..10^6]); # Robert Israel, Dec 19 2014
  • Mathematica
    fQ[n_] := Mod[PowerMod[n, n, n - 3] + 3, n - 3] == 0; Select[Range@ 1000000, fQ] (* Michael De Vlieger, Dec 13 2014; modified by Robert G. Wilson v, Dec 19 2014 *)
  • PARI
    isok(n) = (n != 3) && (Mod(n, n-3)^n  == -3); \\ Michel Marcus, Dec 13 2014

Extensions

More terms from Michel Marcus, Dec 13 2014

A251862 Numbers m such that m + 3 divides m^m - 3.

Original entry on oeis.org

3, 7, 10, 27, 727, 1587, 9838, 758206, 789223, 1018846, 1588126, 1595287, 2387206, 4263586, 9494746, 12697378, 17379860, 21480726, 25439767, 38541526, 44219926, 55561536, 62072326, 64335356, 70032586, 83142466, 85409276
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 10 2014

Keywords

Comments

m such that m+3 divides (-3)^m - 3. - Robert Israel, Dec 14 2014

Examples

			3 is in this sequence because 3 + 3 = 6 divides 3^3 - 3 = 24.

Links

Crossrefs

Cf. ...............Numbers n such that x divides y, where:
...x.....y......k=0.......k=1.......k=2........k=3........
..n-k..n^n-k..A000027...A087156...A242787....A242788......
..n-k..n^n+k..A000027..see below..A249751....A252041......
..n+k..n^n-k..A000027...A004275...A251603..this sequence..
..n+k..n^n+k..A000027...A004273...A213382....A242800......
(For x=n-1 and y=n^n+1, the only terms are 0, 2 and 3. - David L. Harden, Jan 14 2015)

Programs

  • Magma
    [n: n in [2..10000] | Denominator((n^n-3)/(n+3)) eq 1];
  • Maple
    select(t ->((-3) &^ (t) - 3) mod (t+3) = 0, [$1..10^6]); # Robert Israel, Dec 14 2014
  • Mathematica
    a251862[n_] := Select[Range[n], Mod[PowerMod[#, #, # + 3] - 3, # + 3] == 0 &]; a251862[10^6] (* Michael De Vlieger, Dec 14 2014, after Robert G. Wilson v at A252041 *)
  • PARI
    isok(n) = Mod(n, n+3)^n == 3; \\ Michel Marcus, Dec 10 2014
  • Python
    A251862_list = [n for n in range(10**6) if pow(-3, n, n+3) == 3] # Chai Wah Wu, Jan 19 2015
  • Sage
    [n for n in range(10^4) if (n + 3).divides((-3)^n - 3)] # Peter Luschny, Jan 17 2015

Extensions

More terms from Michel Marcus, Dec 10 2014

A357125 Positive integers n such that 2^(n-3) == -1 (mod n).

Original entry on oeis.org

1, 5, 4553, 46777, 82505, 4290773, 4492205, 4976429, 21537833, 21549349, 51127261, 56786089, 60296573, 80837773, 87761789, 94424465, 138644873, 168865001, 221395541, 255881453, 297460453, 305198249, 360306365, 562654205, 635374253, 673867253, 808333573, 1164757553, 1210317349
Offset: 1

Views

Author

Max Alekseyev, Sep 13 2022

Keywords

Comments

Also, odd integers n dividing 2^n + 8.
Some large terms: 5603900696716667005, 446661376165868432471569407934747098747181600670953926245, 1533278864164902082788937853692280620552397221686019535813.

Links

Crossrefs

The odd terms of A245319.
Cf. A006521, A115976, A276967, A251603.

Programs

  • Mathematica
    Select[Range[2155*10^4],PowerMod[2,#-3,#]==#-1&]//Quiet (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Feb 08 2025 *)

A252606 Numbers j such that j + 2 divides 2^j + 2.

Original entry on oeis.org

3, 4, 16, 196, 2836, 4551, 5956, 25936, 46775, 65536, 82503, 540736, 598816, 797476, 1151536, 3704416, 4290771, 4492203, 4976427, 8095984, 11272276, 13362420, 21235696, 21537831, 21549347, 29640832, 31084096, 42913396, 49960912, 51127259, 55137316, 56786087, 60296571, 70254724, 70836676
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 03 2015

Keywords

Comments

Numbers j such that (2^j + 2)/(j + 2) is an integer. Numbers j such that (2^j - j)/(j + 2) is an integer.
From Robert Israel, Apr 09 2015: (Start)
The even members of this sequence (4, 16, 196, 2836, ...) are the numbers 2*k-2 where k>=3 is odd and 4^k == -8 (mod k).
The odd members of this sequence (3, 4551, 46775, 82503, ...) are the numbers k-2 where k>=3 is odd and 2^k == -8 (mod k). (End)
2^m is in this sequence for m = (2, 4, 16, 36, 120, 256, 456, 1296, 2556, ...), with the subsequence m = 2^k, k = (1, 2, 4, 8, 16, ...). - M. F. Hasler, Apr 09 2015

Examples

			3 is in this sequence because (2^3 + 2)/(3 + 2) = 2.

Links

Crossrefs

Cf. A001477, A004273, A004275, A081765, A213382, A251603.

Programs

  • Magma
    [n: n in [0..1200000] | Denominator((2^n+2)/(n+2)) eq 1];
  • Maple
    select(t -> 2 &^t + 2 mod (t + 2) = 0, [$1..10^6]); # Robert Israel, Apr 09 2015
  • Mathematica
    Select[Range[10^6],IntegerQ[(2^#+2)/(#+2)]&] (* Ivan N. Ianakiev, Apr 17 2015 *)
  • PARI
    for(n=1,10^5,if((2^n+2)%(n+2)==0,print1(n,", "))) \\ Derek Orr, Apr 05 2015
  • PARI
    is(n)=Mod(2,n+2)^n==-2 \\ M. F. Hasler, Apr 09 2015
  • Python
    A252606_list = [n for n in range(10**4) if pow(2, n, n+2) == n] # Chai Wah Wu, Apr 16 2015

Extensions

a(17)-a(22) from Tom Edgar, Mar 03 2015
More terms from Chai Wah Wu, Apr 16 2015
Showing 1-4 of 4 results.

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