%I #31 Jun 01 2024 11:56:08
%S 0,0,1,1,0,1,0,1,0,0,1,2,0,1,1,1,0,1,0,1,0,1,1,3,0,0,0,0,0,2,1,3,0,0,
%T 0,0,0,1,0,0,0,1,1,3,0,2,2,4,0,0,0,0,0,1,1,1,0,0,1,3,0,3,1,2,0,0,0,1,
%U 0,0,1,2,0,1,0,0,0,0,1,2,0,1,0,2,0,1,1
%N Number of ways to write n in binary representation as a concatenation of primes.
%H Reinhard Zumkeller, <a href="/A090418/b090418.txt">Table of n, a(n) for n = 0..10000</a> (corrected by _Georg Fischer_, Jan 20 2019)
%F a(A090419(n))=0; a(A090420(n))=1; a(A090421(n))>0;
%F a(A090422(n))=1; a(A090423(n))>1;
%F a(A090424(n)) = n and a(m) <> n for m < A090424(n).
%F a(n) = 0 if a = 0 (mod 4); a(n) = a(floor(n/4)) if a = 2 (mod 4). - _M. F. Hasler_, Apr 21 2015
%e n=23 -> '10111': '10"111'==2"7, '101"11'==5"3 and '10111'==23, therefore a(23)=3.
%o (Haskell)
%o import Data.List (stripPrefix, unfoldr)
%o import Data.Maybe (fromJust)
%o a090418 n = a090418_list !! (n-1)
%o a090418_list = 0 : f 2 where
%o f x = (sum $ map g bpss) : f (x + 1) where
%o g ps | suffix == Nothing = 0
%o | suffix' == [] = 1
%o | last suffix' == 0 = 0
%o | otherwise = a090418 $ fromBits suffix'
%o where suffix' = fromJust suffix
%o suffix = stripPrefix ps $ toBits x
%o bpss = take (fromInteger $ a000720 x) $
%o map (toBits . fromInteger) a000040_list
%o toBits = unfoldr
%o (\u -> if u == 0 then Nothing else Just (mod u 2, div u 2))
%o fromBits = foldr (\b v -> 2 * v + b) 0
%o -- _Reinhard Zumkeller_, Aug 06 2012
%o (PARI) A090418(n)={ while( n>9 && !bittest(n,0), bittest(n,1)||return; n>>=2); n<10 && return(isprime(n)); sum(k=2, #binary(n)-2, if(bittest(n, k-1)&&isprime(n%2^k), A090418(n>>k)),isprime(n))} \\ _M. F. Hasler_, Apr 21 2015
%Y Cf. A004676, A007088.
%Y Cf. A191232, A000040.
%Y Cf. A090421, A090423, A257318.
%K nonn,base
%O 0,12
%A _Reinhard Zumkeller_, Nov 30 2003
%E Thanks to _Alex Ratushnyak_, who found an error in A090423, which was the consequence of errors in this sequence; the program was rewritten and data was recomputed by _Reinhard Zumkeller_, Aug 06 2012
%E Data in b-file double-checked with independent PARI code by _M. F. Hasler_, Apr 21 2015