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Numbers m such that all lengths of runs of digits in base 2 representation of m are primes.
4

%I #21 Jan 28 2018 17:38:19

%S 3,7,12,24,28,31,51,56,96,99,103,115,124,127,199,204,224,227,231,248,

%T 384,387,396,408,412,415,455,460,499,508,775,792,796,799,819,824,896,

%U 899,908,920,924,927,992,995,999,1016,1539,1548,1587,1592,1632,1635,1639,1651,1660

%N Numbers m such that all lengths of runs of digits in base 2 representation of m are primes.

%C Intersection of A226227 and A226228.

%C From _Emeric Deutsch_, Jan 27 2018: (Start)

%C Also the indices of the compositions having only prime parts. For the definition of the index of a composition see A298644. For example, 387 is in the sequence since its binary form is 110000011 and the parts of the composition [2,5,2] are prime numbers. 540 is not in the sequence since its binary form is 1000011100 and not all the parts of the composition [1,4,3,2] are primes.

%C The command c(n) from the Maple program yields the composition having index n. (End)

%H Reinhard Zumkeller, <a href="/A226229/b226229.txt">Table of n, a(n) for n = 1..1000</a>

%p Runs := proc (L) local j, r, i, k: npr: j := 1; r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: npr := proc (s) local q, j: q := 0: for j to nops(s) do if isprime(s[j]) = true then q := q+1 else end if end do end proc: A := {}: for n to 1661 do if npr(c(n)) = nops(c(n)) then A := `union`(A, {n}) else end if end do: A; # most of the Maple program is due to _W. Edwin Clark_. # _Emeric Deutsch_, Jan 27 2018

%t Select[Range@1660, And @@ PrimeQ[Length /@ Split@ IntegerDigits[#, 2]] &] (* _Giovanni Resta_, Jun 01 2013 *)

%o (Haskell)

%o import Data.List (group, genericLength)

%o a226229 n = a226229_list !! (n-1)

%o a226229_list = filter

%o (all (== 1) . map (a010051 . genericLength) . group . a030308_row) [1..]

%o -- _Reinhard Zumkeller_, Jun 05 2013

%Y Cf. A033015, A226227, A226228.

%Y Cf. A005811, A030308, A010051.

%K nonn,base

%O 1,1

%A _Alex Ratushnyak_, May 31 2013