cp's OEIS Frontend

A254753 Composite numbers with only prime proper prefixes and suffixes in base 10.

%I A254753 #21 Nov 29 2018 14:59:55
%S A254753 22,25,27,32,33,35,52,55,57,72,75,77,237,297,537,597,713,717,737,2337,
%T A254753 2397,2937,3113,3173,5937,5997,7197,7337,7397,29397,31373,37937,59397,
%U A254753 73313
%N A254753 Composite numbers with only prime proper prefixes and suffixes in base 10.
%C A254753 A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
%C A254753 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 A254753 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 A254753 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 A254753 6 is not a member because its expansion cannot be sliced in two.
%e A254753 The composite 73313 is a member because (7, 3313, 73, 313, 733, 13, 7331, 3) are all primes.
%t A254753 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 A254753 (PARI) isComposite(n) = (n>2)&&(!isprime(n));
%o A254753 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 A254753 isCompositeSlicingIntoPrimes(n,b=10) = isComposite(n) && slicesIntoPrimes(n,b);
%Y A254753 Cf. A020994, A024770, A024785, A254750, A254751, A254752, A254754.
%K A254753 nonn,base,fini,full
%O A254753 1,1
%A A254753 _Stanislav Sykora_, Feb 15 2015