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Numbers k such that k divides A048678(k).
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%I #33 Jan 10 2022 22:30:10

%S 1,2,4,7,8,14,16,28,31,32,56,62,64,83,112,124,127,128,166,224,248,254,

%T 256,332,397,448,496,508,511,512,664,794,891,896,992,1016,1022,1024,

%U 1163,1328,1588,1782,1792,1984,2032,2044,2047,2048,2326,2656,3176,3441,3564,3584,3968

%N Numbers k such that k divides A048678(k).

%C Apparently includes all powers of 2.

%C All numbers 2^(2k+1)-1 are in this sequence, and if n is in this sequence then so is 2n. - _Charlie Neder_, May 17 2019

%e The first few terms of A048678 are 1,2,5,4,9,10,21,8. 2 is a multiple of 2, 5 isn't a multiple of 3, 4 is a multiple of 4, 9 isn't a multiple of 5, 10 isn't a multiple of 6, 21 is a multiple of 7, etc.

%t Select[Range[4000], Divisible[FromDigits[Flatten[IntegerDigits[#, 2] /. {1 -> {0, 1}}], 2], #] &] (* _Amiram Eldar_, Jul 08 2019 after _Robert G. Wilson v_ at A048678 *)

%o (Haskell) bintodec :: [Integer] -> Integerbintodec = sum . zipWith (*) (iterate (*2) 1) . reverse

%o decomp :: (Integer, [Integer]) -> (Integer, [Integer])decomp (x, ys) = if even x then (x `div` 2, 0:ys) else (x - 1, 1:ys)

%o zeck :: Integer -> Integerzeck n = bintodec (1 : snd (last . takeWhile (\(x, _) -> x > 0) $ iterate decomp (n, [])))

%o output :: [Integer]output = filter (\x -> 0 == zeck x `mod` x) [1..100]

%o main :: IO ()main = print output

%Y Cf. A048678, A083420 (subsequence).

%K nonn,base

%O 1,2

%A _Dan Dart_, May 17 2019