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.
%I A363235 #5 Jun 11 2023 12:29:23
%S A363235 0,1,2,3,4,5,8,9,10,11,16,17,18,19,20,21,22,23,32,33,34,35,36,37,38,
%T A363235 39,64,65,66,67,68,69,70,71,72,73,74,75,128,129,130,131,132,133,134,
%U A363235 135,136,137,138,139,144,145,146,147,148,149,150,151,256,257,258,259,260,261,262,263,264,265,266,267
%N A363235 a(0) = 1; let e be the largest multiplicity such that p^e | a(n); for n>0, a(n) = Sum_{j=1..k} 2^(e(j)-1) where k is the index of the greatest power factor p(k)^e(k) such that p(k-1)^e(k-1) > p(k)^(e(k)+1).
%C A363235 A binary compactification of A363250, this sequence rewrites A363250(n) = Product_{i=1..omega(a(n))} p(i)^e(i) instead as Sum_{i=1..omega(a(n))} e(i)-1.
%C A363235 Not a permutation of nonnegative integers.
%H A363235 Michael De Vlieger, <a href="/A363235/b363235.txt">Table of n, a(n) for n = 0..11210</a> (a(11210) = 2^28.)
%H A363235 Michael De Vlieger, <a href="/A363235/a363235.png">Binary tree</a> indicating natural numbers k in red that appear in this sequence for k = 1..16383.
%e A363235 a(1) = 1 since 2^1 is a product of the smallest primes p(i) whose prime power factors decrease as i increases; Hence a(1) = 2^(e(i)-1) = 1.
%e A363235 a(2) = 2 since we can find no power 3^e with e>=1 that is smaller than 2^1, we increment the exponent of 2 and have 2^2, hence a(2) = 2^(e(i)-1) = 2.
%e A363235 a(3) = 3 since indeed we may multiply 2^2 by 3^1; 2^2 > 3^1, hence Sum_{i=1..2} 2^(e(i)-1) = 2^1 + 2^0 = 2+1 = 3.
%e A363235 Table relating this sequence to A363250.
%e A363235 b(n) = A363250(n), f(n) = A067255(n), g(n) = A272011(n), with the latter two
%e A363235 n b(n) f(b(n)) a(n) g(a(n))
%e A363235 ------------------------------------
%e A363235 1 1 0 0 -
%e A363235 2 2 1 1 0
%e A363235 3 4 2 2 1
%e A363235 4 12 2,1 3 1,0
%e A363235 5 8 3 4 2
%e A363235 6 24 3,1 5 2,0
%e A363235 7 16 4 8 3
%e A363235 8 48 4,1 9 3,0
%e A363235 9 144 4,2 10 3,1
%e A363235 10 720 4,2,1 11 3,1,0
%e A363235 11 32 5 16 4
%e A363235 12 96 5,1 17 4,0
%e A363235 13 288 5,2 18 4,1
%e A363235 14 1440 5,2,1 19 4,1,0
%e A363235 15 864 5,3 20 4,2
%e A363235 16 4320 5,3,1 21 4,2,0
%e A363235 17 21600 5,3,2 22 4,2,1
%e A363235 18 151200 5,3,2,1 23 4,2,1,0
%e A363235 19 64 6 32 5
%e A363235 ...
%e A363235 Therefore, a(18) = 23 = 2^4 + 2^2 + 2^1 + 2^0 since b(18) = 151200 = 2^5 * 3^3 * 5^2 * 7^1.
%e A363235 The sequence is a series of intervals, organized so as to begin with 2^k, that begin as follows:
%e A363235 0
%e A363235 1
%e A363235 2..3
%e A363235 4..5
%e A363235 8..11
%e A363235 16..23
%e A363235 32..39
%e A363235 64..75
%e A363235 128..139 144..151
%e A363235 256..267 272..279
%e A363235 512..523 528..535 544..559
%e A363235 1024..1035 1040..1047 1056..1071
%e A363235 2048..2059 2064..2071 2080..2095 2112..2127
%e A363235 ...
%t A363235 Select[Range[0, 300], AllTrue[Differences@ MapIndexed[Prime[First[#2]]^#1 &, Length[#] - Position[#, 1][[All, 1]] &@ IntegerDigits[#, 2] + 1], # < 0 &] &]
%Y A363235 Cf. A000079, A067255, A272011, A347284, A363063, A363250.
%K A363235 nonn
%O A363235 0,3
%A A363235 _Michael De Vlieger_, Jun 09 2023