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A347287
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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.
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2
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1, 3, 5, 11, 23, 39, 75, 151, 279, 559, 1071, 2127, 4255, 8351, 16687, 33327, 66095, 132191, 263263, 526511, 1052847, 2101423, 4202847, 8405695, 16794303, 33587903, 67175807, 134284671, 268568959, 537004415, 1074006399, 2148012799, 4295496447, 8590992639, 17181985279
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OFFSET
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1,2
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COMMENTS
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Binary compactification of A347285.
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.
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LINKS
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FORMULA
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a(n) = row sum of 2^(m-1) where m are terms in row n of A347285.
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EXAMPLE
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a(1) = 1 since we can find no nonzero exponent e such that 3^e < 2^1; 2^(1 - 1) = 2^0 = 1.
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.
a(3) = 5 since 2^3 > 3^1, hence 2^(3-1) + 2^(1-1) = 2^2 + 2^0 = 4+1 = 5.
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.
----------------------------------------------------
1: 1 -> 1
2: 1 2 -> 3
3: 1 3 -> 5
4: 1 2 4 -> 11
5: 1 2 3 5 -> 23
6: 1 2 3 6 -> 39
7: 1 2 4 7 -> 75
8: 1 2 3 5 8 -> 151
9: 1 2 3 5 9 -> 279
10: 1 2 3 4 6 10 -> 559
11: 1 2 3 4 6 11 -> 1071
12: 1 2 3 4 7 12 -> 2127
...
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MATHEMATICA
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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]
(* Generate 10000 terms from 10000 X 10000 bitmap *)
MapIndexed[FromDigits[Reverse@ #1[[1 ;; First[#2]]], 2] &, ImageData@ Import["https://oeis.org/A347287/a347287_1.png"] /. {0. -> 1, 1. -> 0}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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