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.

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A271314 Largest prime factor of the n-th Jacobsthal number, A001045(n).

Original entry on oeis.org

3, 5, 11, 7, 43, 17, 19, 31, 683, 13, 2731, 127, 331, 257, 43691, 73, 174763, 41, 5419, 683, 2796203, 241, 4051, 8191, 87211, 127, 3033169, 331, 715827883, 65537, 20857, 131071, 86171, 109, 25781083, 524287, 22366891, 61681, 8831418697, 5419, 2932031007403, 2113, 18837001
Offset: 3

Views

Author

Altug Alkan, Apr 03 2016

Keywords

Comments

a(22) = 683 is the first repeated term in this sequence. Note that a(n+2) = A129738(n), for n < 20.

Examples

			a(6) = 7 because A001045(6) = 21 = 3*7.

Links

Crossrefs

Essentially a combination of A005420 and A002587.
Cf. A001045, A006530, A060385, A129738, A264137.

Programs

  • Mathematica
    FactorInteger[#][[-1, 1]] & /@ Take[#, -(Length@ # - 3)] &@ CoefficientList[Series[x/(1 - x - 2 x^2), {x, 0, 45}], x] (* Michael De Vlieger, Apr 04 2016, after Robert G. Wilson v at A001045 *)
  • PARI
    a001045(n) = (2^n - (-1)^n) / 3;
a(n) = vecmax(factor(a001045(n))[,1]);

A291855 Numbers k such that gpf(2^k - 1) - 1 is not divisible by k.

Original entry on oeis.org

28, 52, 68, 92, 124, 156, 172, 244, 260, 308, 327, 340, 348, 356, 380, 396, 404, 428, 436, 500, 508, 516, 532, 580, 612, 644, 660, 684, 696, 724, 732, 748, 764, 780, 796, 820, 836, 908, 940, 980, 996, 1056, 1076, 1124, 1172, 1180
Offset: 1

Views

Author

Thomas Ordowski and Altug Alkan, Sep 04 2017

Keywords

Comments

For a(n) <= 10^3, all terms are divisible by 4 except 327 = 3*109.
Primes p such that 4*p is a term are 7, 13, 17, 23, 31, 43, 61, ...
From Thomas Ordowski, Sep 04 2017: (Start)
If p == +-1 (mod 8) and 2^p - 1 is prime, then 4*p is a term.
Conjecture: 4 * A001153(m) for m > 3 is a subsequence.
Primes q such that q-1 is a term are 29, 53, 157, 173, 349, 397, ... (End)

Examples

			28 is a term because gpf(2^28 - 1) == 15 (mod 28).

Crossrefs

Cf. A000225, A005420, A006530.
Subsequence of A292199.

Extensions

a(10)-a(41) from Giovanni Resta, Sep 04 2017
a(42)-a(46) from Max Alekseyev, Sep 13 2017

A336720 A336719 with duplicates removed.

Original entry on oeis.org

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 4432676798593, 2305843009213693951, 57912614113275649087721, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727, 79638304766856507377778616296087448490695649
Offset: 1

Views

Author

Jeppe Stig Nielsen, Aug 01 2020

Keywords

Comments

Also, record values in A005420.
Mersenne primes (A000668) are a subsequence.
For the corresponding orders of 2 (exponents), see A336721.

Crossrefs

Cf. A000668, A005420, A336719, A336721.

Programs

  • PARI
    re=0;for(k=2,+oo,p=vecmax(factor(2^k-1)[,1]);if(p>re,re=p;print1(re,", ")))

A336721 Values k where A336719(k) increases.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 49, 61, 83, 89, 107, 127, 167, 179, 197, 233, 241, 269, 281, 347, 373, 421, 443, 457, 487, 521, 607, 697, 769, 829, 881, 1049, 1063
Offset: 1

Views

Author

Jeppe Stig Nielsen, Aug 01 2020

Keywords

Comments

Also, values k (record positions) where A005420(k) reaches a new maximum. For the maximum itself, see A336720.
A000043 is a subsequence.

Crossrefs

Cf. A000043, A005420, A336719, A336720.

Programs

  • PARI
    re=0;for(k=2,+oo,p=vecmax(factor(2^k-1)[,1]);if(p>re,re=p;print1(k,", ")))

A337431 Numbers k such that the largest prime factor of 2^k - 1 is greater than the largest prime factor of 2^k + 1.

Original entry on oeis.org

3, 5, 7, 9, 13, 14, 17, 19, 26, 27, 31, 33, 34, 35, 37, 46, 49, 51, 59, 61, 62, 65, 69, 74, 77, 78, 82, 83, 86, 89, 93, 97, 103, 107, 115, 118, 121, 122, 123, 127, 129, 130, 131, 133, 137, 141, 142, 143, 144, 145, 147, 150, 153, 154, 165, 166, 169, 170, 174, 175
Offset: 1

Views

Author

Hugo Pfoertner, Sep 23 2020

Keywords

Crossrefs

Cf. A002587, A005420, A337430 (complement).

Programs

  • PARI
    for(n=2,175,my(p=vecmax(factor(2^n-1)[,1]),q=vecmax(factor(2^n+1)[,1]));if(p>q,print1(n,", ")))

A358699 a(n) is the largest prime factor of 2^(prime(n) - 1) - 1.

Original entry on oeis.org

3, 5, 7, 31, 13, 257, 73, 683, 127, 331, 109, 61681, 5419, 2796203, 8191, 3033169, 1321, 599479, 122921, 38737, 22366891, 8831418697, 2931542417, 22253377, 268501, 131071, 28059810762433, 279073, 54410972897, 77158673929, 145295143558111, 2879347902817, 10052678938039
Offset: 2

Views

Author

Hugo Pfoertner, Nov 27 2022

Keywords

Links

Crossrefs

Cf. A005420, A006093, A006530, A061286, A071243, A086019, A098102.
Subsequence of A005420 and of A274906.

Programs

  • PARI
    forprime (p=3, 140, my(f=factor(2^(p-1)-1)); print1(f[#f[,1],1],", "))
  • Python
    from sympy import primefactors, sieve
def A358699(n): return primefactors(2**(sieve[n]-1)-1)[-1] # Karl-Heinz Hofmann, Nov 28 2022

Formula

a(n) = A006530(A098102(n)). - Michel Marcus, Nov 28 2022
a(n) = A005420(A006093(n)). - Amiram Eldar, Dec 01 2022

A140452 2^(a(n))-1 contains an overpseudoprime divisor.

Original entry on oeis.org

11, 22, 23, 25, 28, 29, 33, 35, 36, 37, 39, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109
Offset: 1

Views

Author

Vladimir Shevelev, Jun 26 2008

Keywords

Comments

If p is a prime then p is in the sequence iff 2^p-1 is a composite number.

Links

Crossrefs

Cf. A141232, A137576, A005420, A049479.

Programs

  • PARI
    f(n) = my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
isopp(n) = (n>1) && !isprime(n) && (n == f((n-1)/2)); \\ A141232
isok(n) = {fordiv(2^n-1, d, if (isopp(d), return (1));); return (0);} \\ Michel Marcus, Dec 09 2018

Extensions

More terms from Michel Marcus, Dec 09 2018

A239638 Numbers n such that the semiprime 2^n-1 is divisible by 2n+1.

Original entry on oeis.org

11, 23, 83, 131, 3359, 130439, 406583
Offset: 1

Views

Author

Zak Seidov, Mar 23 2014

Keywords

Comments

All terms are primes == 5 modulo 6 (A005384 Sophie Germain primes).
a(8) >= 500000. - Max Alekseyev, May 28 2022

Examples

			n = 11, 2^n -1 = 2047 = 23*89,
n = 23, 8388607 = 47*178481,
n = 131, 2722258935367507707706996859454145691647 =  263*10350794431055162386718619237468234569.

Crossrefs

Cf. A000043, A001348, A046051, A005384, A005420, A085724, A049479.

Programs

  • Mathematica
    Select[Range[4000], PrimeQ[2*# + 1] && PowerMod[2, #, 2*# + 1] == 1 &&
PrimeQ[(2^# - 1)/(2*# + 1)] &] (* Giovanni Resta, Mar 23 2014 *)
  • PARI
    is(n)=n%6==5 && Mod(2,2*n+1)^n==1 && isprime(2*n+1) && ispseudoprime((2^n-1)/(2*n+1)) \\ Charles R Greathouse IV, Aug 25 2016
  • Python
    from sympy import isprime, nextprime
A239638_list, p = [], 5
while p < 10**6:
    if (p % 6) == 5:
        n = (p-1)//2
        if pow(2,n,p) == 1 and isprime((2**n-1)//p):
            A239638_list.append(n)
    p = nextprime(p) # Chai Wah Wu, Jun 05 2019

Extensions

a(5)-a(6) from Giovanni Resta, Mar 23 2014
a(7) from Eric Chen, added by Max Alekseyev, May 21 2022

A283461 Second-largest prime factor of 2^n - 1, if composite, or 1 otherwise.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 5, 7, 11, 23, 7, 1, 43, 31, 17, 1, 19, 1, 31, 127, 89, 47, 17, 601, 2731, 73, 113, 1103, 151, 1, 257, 89, 43691, 127, 73, 223, 174763, 8191, 41, 13367, 337, 9719, 683, 631, 178481, 4513, 257, 127, 1801, 11119, 2731
Offset: 2

Views

Author

Charles R Greathouse IV, Mar 08 2017

Keywords

Comments

For clarification: if the largest prime factor occurs more than once, then that prime factor is selected.

Links

Crossrefs

Cf. A005420, A193615, A006530, A000225, A108974.

Programs

  • Mathematica
    a[n_] := If[PrimeQ[2^n-1], 1, Block[{f = FactorInteger[2^n-1]}, If[f[[-1, 2]] == 1, f[[-2, 1]], f[[-1, 1]]]]]; a /@ Range[2, 52] (* Giovanni Resta, Mar 08 2017 *)
  • PARI
    a(n)=my(f=factor(2^n-1),t=#f~); if(f[t,2]>1, f[t,1], if(t>1, f[t-1,1], 1))

Formula

a(n) = A006530(A000225(n)/A005420(n)).

A337430 Numbers k such that the largest prime factor of 2^k - 1 is less than the largest prime factor of 2^k + 1.

Original entry on oeis.org

2, 4, 6, 8, 10, 11, 12, 15, 16, 18, 20, 21, 22, 23, 24, 25, 28, 29, 30, 32, 36, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 50, 52, 53, 54, 55, 56, 57, 58, 60, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 79, 80, 81, 84, 85, 87, 88, 90, 91, 92, 94, 95, 96, 98, 99, 100
Offset: 1

Views

Author

Hugo Pfoertner, Sep 23 2020

Keywords

Crossrefs

Cf. A002587, A005420, A337431 (complement).

Programs

  • PARI
    for(n=2,100,my(p=vecmax(factor(2^n-1)[,1]),q=vecmax(factor(2^n+1)[,1]));if(p
Previous Showing 21-30 of 33 results. Next

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