A257318 -id:A257318 - cp's OEIS frontend

A090418 Number of ways to write n in binary representation as a concatenation of primes.

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, 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, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 1, 2, 0, 1, 0, 2, 0, 1, 1

Offset: 0

Reinhard Zumkeller, Nov 30 2003

			n=23 -> '10111': '10"111'==2"7, '101"11'==5"3 and '10111'==23, therefore a(23)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (corrected by _Georg Fischer_, Jan 20 2019)

Cf. A004676, A007088.
Cf. A191232, A000040.
Cf. A090421, A090423, A257318.

import Data.List (stripPrefix, unfoldr)
import Data.Maybe (fromJust)
a090418 n = a090418_list !! (n-1)
a090418_list = 0 : f 2 where
   f x = (sum $ map g bpss) : f (x + 1) where
     g ps | suffix == Nothing = 0
          | suffix' == []     = 1
          | last suffix' == 0 = 0
          | otherwise         = a090418 $ fromBits suffix'
          where suffix' = fromJust suffix
                suffix = stripPrefix ps $ toBits x
     bpss = take (fromInteger $ a000720 x) $
                  map (toBits . fromInteger) a000040_list
   toBits = unfoldr
            (\u -> if u == 0 then Nothing else Just (mod u 2, div u 2))
   fromBits = foldr (\b v -> 2 * v + b) 0
-- Reinhard Zumkeller, Aug 06 2012

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

a(A090419(n))=0; a(A090420(n))=1; a(A090421(n))>0;
a(A090422(n))=1; a(A090423(n))>1;
a(A090424(n)) = n and a(m) <> n for m < A090424(n).
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

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

Data in b-file double-checked with independent PARI code by M. F. Hasler, Apr 21 2015

A256872 Numbers whose binary expansion is the concatenation of the binary expansion of two prime numbers in at least two ways.

Original entry on oeis.org

23, 31, 45, 47, 61, 93, 95, 119, 125, 127, 175, 187, 189, 191, 239, 247, 253, 255, 335, 357, 359, 363, 369, 379, 381, 383, 431, 439, 455, 477, 485, 491, 493, 495, 507, 509, 511, 573, 575, 631, 637, 639, 669, 671

Offset: 1

M. F. Hasler, Apr 21 2015

A simplified variant (and subsequence) of A257318 (and A090421) where the concatenation of any number of primes is considered.
The subsequence of numbers which are concatenation of 2 primes in at least 3 ways is (93, 95, 189, 191, 239, 253, 335, 381, 383, 669, ...).
All terms are odd. Indeed, if an even number n > 2 is concatenation of two primes (in binary), then it is of the form 'n' = 'floor(n/4)''2' (where 'x' is x in binary), and there is no other possible decomposition.

			23 = 10111[2] = (10[2])(111[2]) = (101[2])(11[2]) which is (2)(7) resp. (5)(3).

Cf. A090418, A090421, A090423, A257318.

is(n,c=2)={for(i=2,#binary(n)-2,bittest(n,i-1)&&isprime(n>>i)&&isprime(n%2^i)&&!c--&&return(1))}

A090418(a(n)) >= 2. (Necessary but not sufficient condition. This actually characterizes elements of A257318. For example, all terms of A090423 satisfy this but many of them are not terms of this sequence.)

cp's OEIS Frontend

A090418 Number of ways to write n in binary representation as a concatenation of primes.

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