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%I #10 Sep 02 2021 01:51:52
%S 1,3,5,11,23,39,75,151,279,559,1071,2127,4255,8351,16687,33327,66095,
%T 132191,263263,526511,1052847,2101423,4202847,8405695,16794303,
%U 33587903,67175807,134284671,268568959,537004415,1074006399,2148012799,4295496447,8590992639,17181985279
%N a(n) = Sum_{k = 1..m} 2^(e_k-1) where e_k = floor(log_p_k(p_(k-1)^e_(k-1))) such that e_k > 0.
%C Binary compactification of A347285.
%C A bitmap produced by aligning the places of bits plotted for successive terms traces trajectories of the primes p_k as n increases in A347285. (See "little-endian bitmaps", so-named as the least significant bit appears at left.) For example, the rightmost trajectory pertains to p = 2, and moving left, p = 3, p = 5, etc. The trajectory for p_1 = 2 appears as a 45-degree angle since A347285(n,1) = n by definition.
%H Michael De Vlieger, <a href="/A347287/b347287.txt">Table of n, a(n) for n = 1..3322</a>
%H Michael De Vlieger, <a href="/A347287/a347287.png">Little-endian bitmap</a> of a(n) for n=1..512, black = 1 and white = 0.
%H Michael De Vlieger, <a href="/A347287/a347287_1.png">Little-endian bitmap</a> of a(n) for n=1..10000, black = 1 and white = 0.
%F a(n) = row sum of 2^(m-1) where m are terms in row n of A347285.
%e a(1) = 1 since we can find no nonzero exponent e such that 3^e < 2^1; 2^(1 - 1) = 2^0 = 1.
%e a(2) = 3 since 3^1 < 2^2 yet 3^2 > 2^2. (We assume hereinafter that the powers listed are the largest possible smaller than the immediately previous term.) 2^(2-1) + 2^(1-1) = 2^1+2^0 = 2+1 = 3.
%e a(3) = 5 since 2^3 > 3^1, hence 2^(3-1) + 2^(1-1) = 2^2 + 2^0 = 4+1 = 5.
%e a(4) = 11 since 2^4 > 3^2 > 5^1, thus 2^(4-1) + 2^(2-1) + 2(1-1) = 8+2+1 = 11, etc.
%e n Row n of A347285 (reversed) a(n)
%e ----------------------------------------------------
%e 1: 1 -> 1
%e 2: 1 2 -> 3
%e 3: 1 3 -> 5
%e 4: 1 2 4 -> 11
%e 5: 1 2 3 5 -> 23
%e 6: 1 2 3 6 -> 39
%e 7: 1 2 4 7 -> 75
%e 8: 1 2 3 5 8 -> 151
%e 9: 1 2 3 5 9 -> 279
%e 10: 1 2 3 4 6 10 -> 559
%e 11: 1 2 3 4 6 11 -> 1071
%e 12: 1 2 3 4 7 12 -> 2127
%e ...
%t Array[Total[2^(-1 + NestWhile[Block[{p = Prime[#2]}, Append[#1, {p^#, #} &@ Floor@ Log[p, #1[[-1, 1]]]]] & @@ {#, Length@ # + 1} &, {{2^#, #}}, #[[-1, -1]] > 1 &][[All, -1]])] &, 35]
%t (* Generate 10000 terms from 10000 X 10000 bitmap *)
%t MapIndexed[FromDigits[Reverse@ #1[[1 ;; First[#2]]], 2] &, ImageData@ Import["https://oeis.org/A347287/a347287_1.png"] /. {0. -> 1, 1. -> 0}]
%Y Cf. A000079, A000961, A347285.
%K nonn,easy
%O 1,2
%A _Michael De Vlieger_, Aug 30 2021
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