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.

A077315 Primes of the form 2^r * 11^s - 1.

Original entry on oeis.org

3, 7, 31, 43, 127, 241, 967, 5323, 8191, 117127, 131071, 524287, 7496191, 10307263, 77948683, 253755391, 428717761, 738197503, 1714871047, 2147483647, 16240345087, 27437936767, 42218553343, 1965081755647, 2414538435583, 7024111812607, 7860327022591, 16630113370111
Offset: 1

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Links

Crossrefs

Cf. A023509, A077313, A077314, A173062.

Programs

  • PARI
    lista(nn) = {my(q=1/2, p, w=List([])); for(r=0, logint(nn, 2), q=2*q; p=q/11; for(s=0, logint(nn\q, 11), p=11*p; if(ispseudoprime(p-1), listput(w, p-1)))); Set(w); } \\ Jinyuan Wang, Feb 23 2020

Extensions

More terms from Ray Chandler, Aug 02 2003
a(26)-a(28) from Jinyuan Wang, Feb 23 2020

A173062 Primes of the form 2^r * 13^s - 1.

Original entry on oeis.org

3, 7, 31, 103, 127, 337, 1663, 5407, 8191, 131071, 346111, 524287, 2970343, 3655807, 22151167, 109051903, 617831551, 1631461441, 2007952543, 2147483647, 32127240703, 194664464383, 275716983697, 958348132351, 1357375919743, 1670616516607, 49834102882303, 57349132609183
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 09 2010

Keywords

Comments

s = 0 is "trivial" case of Mersenne primes: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...
Mersenne prime exponents r: 2, 3, 5, 7, 13, 17, 19, 31, ...
Necessarily r odd as for r = 2*k and p a prime of form 6*n+1: 2^(2*k) * p^j - 1 a multiple of 3.
Proof by induction with 2^2 * p^1 - 1 = 4*(6*n+1) - 1 = 3*(8*n+1), 2^2(k+1) * p^j - 1 = 4* (2^k * p^j - 1) + 3.
No prime in case i = j = k (k>1) as a^k - 1 has divisor a - 1.

Examples

			2^2*13^0 - 1 = 3 = prime(2) => a(1).
2^3*13^1 - 1 = 103 = prime(27) => a(4).
2^7*13^9 - 1 = 1357375919743 = prime(50467169414) => a(25).
list of (r,s) pairs: (2,0), (3,0), (5,0), (3,1), (7,0), (1,2), (7,1), (5,2), (13,0), (17,0), (11,2), (19,0), (3,5), (7,4), (17,2), (23,1), (7,6), (1,8), (5,7), (31,0), (9,7), (19,5), (1,10), (25,4), (7,9), (11,8), (27,5), (5,11), (25,6), (19,8), (13,10), (3,13), (29,7), (5,14), (39,5), (15,13), (5,16), ...

References

  • Peter Bundschuh, Einfuehrung in die Zahlentheorie, Springer-Verlag GmbH Berlin, 2002.
  • Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications, 2005.
  • Paulo Ribenboim, Wilfrid Keller, and Joerg Richstein, Die Welt der Primzahlen, Springer-Verlag GmbH Berlin, 2006.

Links

Crossrefs

Cf. A000668, A005105, A077313, A077314, A077315.

Programs

  • PARI
    lista(nn) = {my(q=1/2, p, w=List([])); for(r=0, logint(nn, 2), q=2*q; p=q/13; for(s=0, logint(nn\q, 13), p=13*p; if(ispseudoprime(p-1), listput(w, p-1)))); Set(w); } \\ Jinyuan Wang, Feb 23 2020
  • Python
    from itertools import count, islice
from sympy import integer_log, isprime
def A173062_gen(): # generator of terms
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(n):
        def f(x): return n+x-sum(((x+1)//13**i).bit_length() for i in range(integer_log(x+1,13)[0]+1))
        return bisection(f,n-1,n-1)
    return filter(lambda n:isprime(n), map(g,count(1)))
A173062_list = list(islice(A173062_gen(),30)) # Chai Wah Wu, Mar 31 2025

Extensions

a(26)-a(28) from Jinyuan Wang, Feb 23 2020

A086983 Primes of the form 2^r*p^s - 1, where p is an odd prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 43, 47, 53, 61, 67, 71, 73, 79, 97, 103, 107, 127, 151, 157, 163, 191, 193, 199, 211, 223, 241, 271, 277, 283, 313, 331, 337, 367, 383, 397, 421, 431, 457, 463, 487, 499, 523, 541, 547, 577, 607, 613, 631, 647, 661, 673
Offset: 1

Views

Author

Ray Chandler, Aug 02 2003

Keywords

Comments

Primes p such that p+1 has at most one odd prime divisor.

Links

Crossrefs

Cf. A005105, A077497, A077498, A077499, A077500.
Cf. also A005109, A077313, A077314, A077315.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
Primes:= select(isprime, [$3..(N+1)/2]):
sort(convert(select(isprime, {2,seq(seq(seq(2^r*p^s-1, r = 1 .. ilog2((N+1)/p^s)),s=0..floor(log[p]((N+1)/2))),p=Primes)}),list)); # Robert Israel, Jun 13 2018

A292890 Primes of the form 2^r * 17^s - 1.

Original entry on oeis.org

3, 7, 31, 67, 127, 271, 577, 1087, 2311, 8191, 78607, 131071, 524287, 1114111, 2367487, 2672671, 17825791, 42762751, 90870847, 606076927, 2147483647, 5151653887, 5815734271, 9697230847, 329705848831, 474351505987, 700624928767, 892896952447, 1168231104511, 2482491097087
Offset: 1

Views

Author

Muniru A Asiru, Sep 26 2017

Keywords

Comments

Primes of the forms 2^r * b^s - 1 where b = 1, 5, 7, 11, 13 are A000668 (Mersenne prime exponents), A077313, A077314, A077315 and A173062. When b = 3 we get A005105 with initial term 2.
For n > 1, all terms are congruent to 1 (mod 3).
Also, these are prime numbers p for which (34^p)/(p+1) is an integer.

Examples

			With n = 1, a(1) = 2^2 * 17^0 - 1 = 3.
With n = 4, a(4) = 2^2 * 17^1 - 1 = 67.
list of (r, s): (2, 0), (3, 0), (5, 0), (2, 1), (3, 1), (7, 0), (4, 1), (1, 2), (6, 1), (3, 2), (13, 0), (4, 3), (17, 0), (19, 0), (16, 1), (13, 2), (5, 4), (20, 1), (9, 4), (6, 5).

Crossrefs

Cf. Sequences of primes of the forms 2^n * q^s - 1: A000668 (q = 1), A005105 (q = 3), A077313 (q = 5), A077314 (q = 7), A077315 (q = 11), A173062 (q = 13).

Programs

  • GAP
    K:=10^7+1;; # to get all terms <= K.
A:=Filtered(Filtered([1..K], i->i mod 3=1),IsPrime);;    I:=[17];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A292890:=Concatenation([3],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]));
  • PARI
    isok(p) = isprime(p) && (denominator((34^p)/(p+1)) == 1); \\ Michel Marcus, Sep 27 2017

Extensions

More terms from Jinyuan Wang, Feb 23 2020
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.