A347287 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.

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

Offset: 1

Author

Michael De Vlieger, Aug 30 2021

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
n      Row n of A347285 (reversed)              a(n)
----------------------------------------------------
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
...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..3322
• Michael De Vlieger, Little-endian bitmap of a(n) for n=1..512, black = 1 and white = 0.
• Michael De Vlieger, Little-endian bitmap of a(n) for n=1..10000, black = 1 and white = 0.

Crossrefs

Cf. A000079, A000961, A347285.

Programs

Mathematica

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}]

Formula

a(n) = row sum of 2^(m-1) where m are terms in row n of A347285.