A254753 - OEIS

%I #21 Nov 29 2018 14:59:55

%S 22,25,27,32,33,35,52,55,57,72,75,77,237,297,537,597,713,717,737,2337,

%T 2397,2937,3113,3173,5937,5997,7197,7337,7397,29397,31373,37937,59397,

%U 73313

%N Composite numbers with only prime proper prefixes and suffixes in base 10.

%C A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.

%C Alternative definition: Slicing the decimal expansion of a composite a(n) in any way into two nonempty parts, each part represents a prime number.

%C This sequence is a subset of A254751. Every proper prefix of each member a(n) is a member of A024770, and every proper suffix is a member of A024785. Since the latter are finite sequences, a(n) is also a finite sequence. It has 34 members, the largest of which is the composite number 73313.

%C Should one change the definition to 'prime numbers such that, in base 10, all their proper prefixes and suffixes represent primes', the result would be the sequence A020994.

%e 6 is not a member because its expansion cannot be sliced in two.

%e The composite 73313 is a member because (7, 3313, 73, 313, 733, 13, 7331, 3) are all primes.

%t apQ[n_]:=Module[{idn=IntegerDigits[n],c1,c2},c1=FromDigits/@ Table[ Take[ idn,k],{k,Length[idn]-1}];c2=FromDigits/@Table[Take[idn,k],{k,-(Length[ idn]-1), -1}]; AllTrue[ Join[c1,c2],PrimeQ]]; Select[Range[ 10,80000], CompositeQ[#] && apQ[#]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Nov 29 2018 *)

%o (PARI) isComposite(n) = (n>2)&&(!isprime(n));

%o slicesIntoPrimes(n,b=10) = {my(k=b);if(n<b,return(0););while(n\k>0,if(!isprime(n\k)||!isprime(n%k),return(0););k*=b;);1;}

%o isCompositeSlicingIntoPrimes(n,b=10) = isComposite(n) && slicesIntoPrimes(n,b);

%Y Cf. A020994, A024770, A024785, A254750, A254751, A254752, A254754.

%K nonn,base,fini,full

%O 1,1

%A _Stanislav Sykora_, Feb 15 2015