A022330 - OEIS

%I #78 Sep 16 2024 12:46:40

%S 1,3,7,12,19,27,37,49,62,77,93,111,131,152,175,199,225,252,281,312,

%T 344,378,413,450,489,529,571,614,659,705,753,803,854,907,961,1017,

%U 1075,1134,1195,1257,1321,1386,1453,1522,1592,1664,1737,1812,1889,1967,2047,2128

%N Index of 3^n within sequence of numbers of form 2^i*3^j (A003586).

%C a(1000)=793775, a(10000)=79261054, a(100000)=7924941755, a(1000000)=792482542841.

%H Zak Seidov, <a href="/A022330/b022330.txt">Table of n, a(n) for n = 0..10000</a> (terms for n = 0..1000 from Charles R Greathouse IV).

%H N. Carey, <a href="http://arxiv.org/abs/1303.0888">Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet</a>, arXiv preprint arXiv:1303.0888 [math.CO], 2013.

%H N. Carey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Carey/carey6.html">Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet</a>, J. Int. Seq. 16 (2013) #13.3.4.

%F a(n) = A071521(A000244(n)); A003586(a(n)) = A000244(n). - _Reinhard Zumkeller_, May 09 2006

%F a(n) ~ kn^2 with k = log(3)/log(4) = 0.792.... More exact asymptotics? - _Zak Seidov_, Dec 22 2011

%F a(n+1) = a(n) + A020914(n+1). - _Ruud H.G. van Tol_, Nov 25 2022

%F kn^2 + kn + 1 <= a(n) <= kn^2 + (k+1)n + 1, so a(n) = kn^2 + O(n) with k = log(3)/log(4). The law of the iterated logarithm suggests that a better error term might be possible. - _Charles R Greathouse IV_, Nov 28 2022

%t c[0] = 1; c[n_] := 1 + Sum[Ceiling[j*Log[2, 3]], {j, n}]; Table[c[i], {i, 0, 51}] (* _Norman Carey_, Jun 13 2012 *)

%o (PARI) listsm(lim)=my(v=List(),N); for(n=0,log(lim)\log(3),N=3^n; while(N<=lim,listput(v,N);N<<=1)); v=Vec(v); vecsort(v)

%o list(lim)=my(v=listsm(3^floor(lim)));vector(floor(lim+1),i,setsearch(v,3^(i-1))) \\ _Charles R Greathouse IV_, Aug 19 2011

%o (PARI) a(n)=sum(k=0,n, logint(3^k,2))+n+1 \\ _Charles R Greathouse IV_, Nov 22 2022

%o (Python)

%o def A022330(n): return sum((3**i).bit_length() for i in range(n+1)) # _Chai Wah Wu_, Sep 16 2024

%Y Cf. A022331, A020914 (first differences).

%Y Cf. A000244, A003586, A071521.

%K nonn

%O 0,2

%A _Clark Kimberling_