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.

A247165 Numbers m such that m^2 + 1 divides 2^m - 1.

Original entry on oeis.org

0, 16, 256, 8208, 65536, 649800, 1382400, 4294967296
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 30 2014

Keywords

Comments

Contains 2^(2^k) = A001146(k) for k >= 2. - Robert Israel, Dec 02 2014
a(9) > 10^12. - Hiroaki Yamanouchi, Sep 16 2018
For each n, a(n)^2 + 1 belongs to A176997, and thus a(n) belongs to either A005574 or A135590. - Max Alekseyev, Feb 08 2024

Examples

			0 is in this sequence because 0^2 + 1 = 1 divides 2^0 - 1 = 1.

Crossrefs

Cf. A247219 (n^2 - 1 divides 2^n - 1), A247220 (n^2 + 1 divides 2^n + 1).
Cf. A005574, A135590, A176997.

Programs

  • Magma
    [n: n in [1..100000] | Denominator((2^n-1)/(n^2+1)) eq 1];
  • Maple
    select(n -> (2 &^ n - 1) mod (n^2 + 1) = 0, [$1..10^6]); # Robert Israel, Dec 02 2014
  • Mathematica
    a247165[n_Integer] := Select[Range[0, n], Divisible[2^# - 1, #^2 + 1] &]; a247165[1500000] (* Michael De Vlieger, Nov 30 2014 *)
  • PARI
    for(n=0,10^9,if(Mod(2,n^2+1)^n==+1,print1(n,", "))); \\ Joerg Arndt, Nov 30 2014
  • Python
    A247165_list = [n for n in range(10**6) if n == 0 or pow(2,n,n*n+1) == 1]
# Chai Wah Wu, Dec 04 2014

Extensions

a(8) from Chai Wah Wu, Dec 04 2014
Edited by Jon E. Schoenfield, Dec 06 2014

A247221 Numbers k such that 2*k^2 + 1 divides 2^k + 1.

Original entry on oeis.org

0, 1, 67653, 2124804
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 30 2014

Keywords

Comments

Numbers k such that (2^k + 1)/(2*k^2 + 1) is an integer.
a(5) > 2*10^10. - Chai Wah Wu, Dec 07 2014

Crossrefs

Cf. A247205, A247220.

Programs

  • Magma
    [n: n in [1..300000] | Denominator((2^n+1)/(2*n^2+1)) eq 1];
  • Mathematica
    a247221[n_Integer] := Select[Range[n], Divisible[2^# + 1, 2*#^2 + 1] &]; a247221[2500000] (* Michael De Vlieger, Nov 30 2014 *)
  • PARI
    for(n=0,10^9,if(Mod(2,2*n^2+1)^n==-1,print1(n,", "))); \\ Joerg Arndt, Nov 30 2014
  • Python
    A247221_list = [n for n in range(10**6) if pow(2,n,2*n*n+1) == 2*n*n]
# Chai Wah Wu, Dec 07 2014

A273870 Numbers m such that 4^(m-1) == 1 (mod (m-1)^2 + 1).

Original entry on oeis.org

1, 3, 5, 17, 217, 257, 387, 8209, 20137, 37025, 59141, 65537, 283801, 649801, 1382401, 373164545, 535019101, 2453039425, 4294967297
Offset: 1

Views

Author

Jaroslav Krizek, Jun 01 2016

Keywords

Comments

Also, numbers m such that (4^k)^(m-1) == 1 (mod (m-1)^2 + 1) for all k >= 0.
a(20) > 2*10^12, if it exists. - Giovanni Resta, Feb 26 2020

Examples

			5 is a term because 4^(5-1) == 1 (mod (5-1)^2+1), i.e., 255 == 0 (mod 17).

Crossrefs

Prime terms are in A273871.
Contains A000215 (Fermat numbers) as subsequence.
Contains 1 + A247220 as subsequence.
Cf. A019434, A273999.

Programs

  • Magma
    [n: n in [1..100000] | (4^(n-1)-1) mod ((n-1)^2+1) eq 0];
  • PARI
    isok(n) = Mod(4, (n-1)^2+1)^(n-1) == 1; \\ Michel Marcus, Jun 02 2016

Formula

a(n) = sqrt(A273999(n)-1) + 1. - Jinyuan Wang, Feb 24 2020

Extensions

a(14)-a(15) from Michel Marcus, Jun 02 2016
Edited by Max Alekseyev, Apr 30 2018
a(16)-a(19) from Jinyuan Wang, Feb 24 2020

A319216 Numbers k such that k^2 + 1 divides 2^k + 2.

Original entry on oeis.org

0, 1, 3, 15, 79, 511, 4095, 6735, 65535, 2097151, 16777215, 75955411, 68719476735, 137438953471
Offset: 1

Views

Author

Altug Alkan, Sep 13 2018, following a suggestion from Max Alekseyev

Keywords

Comments

Numbers t such that 2^t-1 is a term are 0, 1, 2, 4, 9, 12, 16, 21, ...
Primes p such that 2^((p^2-1)/2)-1 is a term are 3, 5, 7, 13, 19, 29, 37, 43, 47, 61, 71, 89, 103, 107, 127, 131, 139, 173, 191, 211, ...(cf. A062326).
a(14) > 10^11. - Hiroaki Yamanouchi, Sep 14 2018

Crossrefs

Cf. A015921, A062326, A097958, A244673, A247220.

Programs

  • PARI
    isok(n)=Mod(2, n^2+1)^n==-2;

Extensions

a(13) from Hiroaki Yamanouchi, Sep 14 2018
a(14) from Giovanni Resta, Sep 17 2018

A319233 Numbers k such that k^2 + 1 divides 2^k + 4.

Original entry on oeis.org

0, 1, 8, 28, 32, 128, 2048, 8192, 23948, 131072, 524288, 8388608, 536870912, 2147483648, 137438953472
Offset: 1

Views

Author

Altug Alkan, Sep 14 2018

Keywords

Comments

This sequence corresponds to numbers k such that k^2 + 1 divides 2^k + 2^m where m = 2 (A247220 (m = 0), A319216 (m = 1)).
a(16) > 10^12. - Hiroaki Yamanouchi, Sep 17 2018

Examples

			32 = 2^5 is a term since (2^(2^5) + 2^2)/((2^5)^2 + 1) = 2^22 - 2^12 + 2^2.

Crossrefs

Cf. A034785, A247220, A319216.

Programs

  • PARI
    isok(n)=Mod(2, n^2+1)^n==-4;

Extensions

a(15) from Hiroaki Yamanouchi, Sep 17 2018

A319245 Numbers k such that k^2 + 1 divides 2^k + 8.

Original entry on oeis.org

0, 1, 17, 37, 77, 197, 513, 993, 1837, 2617, 2637, 4097, 5437, 65537, 261633, 364137, 437837, 2097153, 16777217, 32761917, 54644032237, 68719476737, 137438953473, 1099511627777
Offset: 1

Views

Author

Altug Alkan, Sep 15 2018

Keywords

Comments

Prime terms are 17, 37, 197, 2617, 5437, 65537, 437837, ...
Numbers t such that 2^t + 1 is a term are 4, 9, 12, 16, 21, 24, 36, 37, 40, 45, 49, 52, 57, 64, 69, 76, 84, 96, ...

Crossrefs

Cf. A247220, A319216, A319233.

Programs

  • Mathematica
    Select[Range[0, 9999], Divisible[2^# + 8, #^2 + 1] &] (* Alonso del Arte, Sep 16 2018 *)
  • PARI
    isok(n)=Mod(2, n^2+1)^n==-8;

Extensions

a(21)-a(24) from Hiroaki Yamanouchi, Sep 16 2018
Showing 1-6 of 6 results.

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