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 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). 1
 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, 39, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 128, 129, 130, 131, 132, 133, 134, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. Not a permutation of nonnegative integers. LINKS Michael De Vlieger, Table of n, a(n) for n = 0..11210 (a(11210) = 2^28.) Michael De Vlieger, Binary tree indicating natural numbers k in red that appear in this sequence for k = 1..16383. EXAMPLE 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. 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. 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. Table relating this sequence to A363250. b(n) = A363250(n), f(n) = A067255(n), g(n) = A272011(n), with the latter two n b(n) f(b(n)) a(n) g(a(n)) ------------------------------------ 1 1 0 0 - 2 2 1 1 0 3 4 2 2 1 4 12 2,1 3 1,0 5 8 3 4 2 6 24 3,1 5 2,0 7 16 4 8 3 8 48 4,1 9 3,0 9 144 4,2 10 3,1 10 720 4,2,1 11 3,1,0 11 32 5 16 4 12 96 5,1 17 4,0 13 288 5,2 18 4,1 14 1440 5,2,1 19 4,1,0 15 864 5,3 20 4,2 16 4320 5,3,1 21 4,2,0 17 21600 5,3,2 22 4,2,1 18 151200 5,3,2,1 23 4,2,1,0 19 64 6 32 5 ... Therefore, a(18) = 23 = 2^4 + 2^2 + 2^1 + 2^0 since b(18) = 151200 = 2^5 * 3^3 * 5^2 * 7^1. The sequence is a series of intervals, organized so as to begin with 2^k, that begin as follows: 0 1 2..3 4..5 8..11 16..23 32..39 64..75 128..139 144..151 256..267 272..279 512..523 528..535 544..559 1024..1035 1040..1047 1056..1071 2048..2059 2064..2071 2080..2095 2112..2127 ... MATHEMATICA Select[Range[0, 300], AllTrue[Differences@ MapIndexed[Prime[First[#2]]^#1 &, Length[#] - Position[#, 1][[All, 1]] &@ IntegerDigits[#, 2] + 1], # < 0 &] &] CROSSREFS Cf. A000079, A067255, A272011, A347284, A363063, A363250. Sequence in context: A032877 A032844 A023775 * A329296 A356713 A032867 Adjacent sequences: A363232 A363233 A363234 * A363236 A363237 A363238 KEYWORD nonn AUTHOR Michael De Vlieger, Jun 09 2023 STATUS approved

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Last modified March 4 15:47 EST 2024. Contains 370532 sequences. (Running on oeis4.)