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
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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
A173236
Primes of the form 2^r * 13^s + 1.
Original entry on oeis.org
2, 3, 5, 17, 53, 257, 677, 3329, 13313, 35153, 65537, 2768897, 13631489, 2303721473, 3489660929, 4942652417, 11341398017, 10859007357953, 1594691292233729, 31403151600910337, 310144109150467073, 578220423796228097
Offset: 1
Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 13 2010
2^0*13^0 + 1 = 2 = prime(1) => a(1).
2^10*13^1 + 1 = 13313 = prime(1581) => a(9).
list of (r,s): (0,0), (1,0), (2,0), (4,0), (2,1), (8,0), (2,2), (8,1), (10,1), (4,3), (16,0), (14,2), (20,1), (20,3), (28,1), (10,6), (26,2), (10,9), (32,5), (40,4), (10,13), (22,10), (32,8), (48,4), (20,13), (2,18), (28,11), (50,6).
- Emil Artin, Galoissche Theorie, Verlag Harri Deutsch, Zürich, 1973.
- 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.
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K:=10^7;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;
B:=List(A,i->Factors(i-1));;
C:=[];; for i in B do if Elements(i)=[2] or Elements(i)=[2,13] then Add(C,Position(B,i)); fi; od;
A173236:=Concatenation([2],List(C,i->A[i])); # Muniru A Asiru, Sep 10 2017
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from itertools import count, islice
from sympy import isprime, integer_log
def A173236_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)))
A173236_list = list(islice(A173236_gen(),30)) # Chai Wah Wu, Mar 31 2025
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
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).
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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]));
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isok(p) = isprime(p) && (denominator((34^p)/(p+1)) == 1); \\ Michel Marcus, Sep 27 2017
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