cp's OEIS Frontend

A069353 Numbers of form 2^i*3^j - 1 with i, j >= 0.

%I A069353 #20 Apr 22 2025 03:46:52
%S A069353 0,1,2,3,5,7,8,11,15,17,23,26,31,35,47,53,63,71,80,95,107,127,143,161,
%T A069353 191,215,242,255,287,323,383,431,485,511,575,647,728,767,863,971,1023,
%U A069353 1151,1295,1457,1535,1727,1943,2047,2186,2303,2591,2915,3071,3455,3887
%N A069353 Numbers of form 2^i*3^j - 1 with i, j >= 0.
%C A069353 Are there infinitely many primes in this sequence? See A005105.
%C A069353 If m is a term then also 2*m + 1 and 3*m + 2.
%H A069353 Graham Everest, Peter Rogers, and Thomas Ward, <a href="https://doi.org/10.1007/3-540-45455-1_8">A higher-rank Mersenne problem</a>, Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7-12, 2002 Proceedings 5, Lect. Notes Computer Sci. 2369, Springer Berlin Heidelberg, 2002, pp. 95-107.
%F A069353 a(n) = A003586(n)-1.
%t A069353 With[{max = 4000}, Sort[Flatten[Table[2^i*3^j - 1, {i, 0, Log2[max]}, {j, 0, Log[3, max/2^i]}]]]] (* _Amiram Eldar_, Jul 13 2023 *)
%o A069353 (Python)
%o A069353 from sympy import integer_log
%o A069353 def A069353(n):
%o A069353     def bisection(f,kmin=0,kmax=1):
%o A069353         while f(kmax) > kmax: kmax <<= 1
%o A069353         kmin = kmax >> 1
%o A069353         while kmax-kmin > 1:
%o A069353             kmid = kmax+kmin>>1
%o A069353             if f(kmid) <= kmid:
%o A069353                 kmax = kmid
%o A069353             else:
%o A069353                 kmin = kmid
%o A069353         return kmax
%o A069353     def f(x): return n+x-sum(((x+1)//3**i).bit_length() for i in range(integer_log(x+1,3)[0]+1))
%o A069353     return bisection(f,n-1,n-1) # _Chai Wah Wu_, Mar 31 2025
%Y A069353 Cf. A003586, A055600, A069355, A005105.
%K A069353 nonn
%O A069353 1,3
%A A069353 _Reinhard Zumkeller_, Mar 18 2002