cp's OEIS Frontend

A319293 Numbers of the form 27^i*(9*j +- 1).

%I A319293 #15 Feb 15 2025 02:14:33
%S A319293 1,8,10,17,19,26,27,28,35,37,44,46,53,55,62,64,71,73,80,82,89,91,98,
%T A319293 100,107,109,116,118,125,127,134,136,143,145,152,154,161,163,170,172,
%U A319293 179,181,188,190,197,199,206,208,215,216,217,224,226,233,235,242,244
%N A319293 Numbers of the form 27^i*(9*j +- 1).
%C A319293 {+-a(n)} gives all nonzero cubes modulo all powers of 3, that is, cubes over 3-adic integers. So this sequence is closed under multiplication.
%H A319293 Jianing Song, <a href="/A319293/b319293.txt">Table of n, a(n) for n = 1..9231</a> (all terms <= 40000)
%F A319293 a(n) = 13*n/3 + O(log(n)).
%o A319293 (PARI) isA319293(n)= n\27^valuation(n, 27)%9==1||n\27^valuation(n, 27)%9==8
%o A319293 (Python)
%o A319293 from sympy import integer_log
%o A319293 def A319293(n):
%o A319293     def bisection(f,kmin=0,kmax=1):
%o A319293         while f(kmax) > kmax: kmax <<= 1
%o A319293         kmin = kmax >> 1
%o A319293         while kmax-kmin > 1:
%o A319293             kmid = kmax+kmin>>1
%o A319293             if f(kmid) <= kmid:
%o A319293                 kmax = kmid
%o A319293             else:
%o A319293                 kmin = kmid
%o A319293         return kmax
%o A319293     def f(x): return n+x-sum(((m:=x//27**i)-1)//9+(m-8)//9+2 for i in range(integer_log(x,27)[0]+1))
%o A319293     return bisection(f,n,n) # _Chai Wah Wu_, Feb 14 2025
%Y A319293 A056020 is a proper subsequence.
%Y A319293 Perfect powers over 3-adic integers:
%Y A319293 Squares: positive: A055047; negative: A055048 (negated);
%Y A319293 Cubes: this sequence.
%K A319293 nonn
%O A319293 1,2
%A A319293 _Jianing Song_, Sep 16 2018