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 A075158 Prime factorization of n+1 encoded with the run lengths of binary expansion. 17

%I

%S 0,1,2,3,5,4,10,7,6,11,21,8,42,20,9,15,85,12,170,23,22,43,341,16,13,

%T 84,14,40,682,19,1365,31,41,171,18,24,2730,340,86,47,5461,44,10922,87,

%U 17,683,21845,32,26,27,169,168,43690,28,45,80,342,1364,87381,39,174762

%N Prime factorization of n+1 encoded with the run lengths of binary expansion.

%C a(2n) = 1 or 2 mod 4 and a(2n+1) = 0 or 3 mod 4 for all n > 1

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e a(1) = 1 as 2 = 2^1, a(2) = 2 (10 in binary) as 3 = 3^1 * 2^0, a(3) = 3 (11) as 4 = 2^2, a(4) = 5 (101) as 5 = 5^1 * 3^0 * 2^0, a(5) = 4 (100) as 6 = 3^1 * 2^1, a(8) = 6 (110) as 9 = 3^2 * 2^0, a(11) = 8 (1000) as 12 = 3^1 * 2^2, a(89) = 35 (100011) as 90 = 5^1 * 3^2 * 2^1, a(90) = 90 (1011010) as 91 = 13^1 * 11^0 * 7^1 * 5^0 * 3^0 * 2^0.

%e The binary expansion of a(n) begins from the left with as many 1's as is the exponent of the largest prime present in the factorization of n+1 and from then on follows runs of ej+1 zeros and ones alternatively, where ej are the corresponding exponents of the successively lesser primes (0 if that prime does not divide n+1).

%o import Data.List (elemIndex); import Data.Maybe (fromJust)

%o a075158 = fromJust . (`elemIndex` a075157_list)

%o -- _Reinhard Zumkeller_, Aug 04 2014

%Y Inverse of A075157. a(n) = A075160(n+1)-1. a(A006093(n)) = A000975(n). Cf. A059884.

%K nonn

%O 0,3

%A _Antti Karttunen_, Sep 13 2002

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Last modified July 14 10:29 EDT 2020. Contains 335721 sequences. (Running on oeis4.)