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

22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 237, 297, 537, 597, 713, 717, 737, 2337, 2397, 2937, 3113, 3173, 5937, 5997, 7197, 7337, 7397, 29397, 31373, 37937, 59397, 73313

Offset: 1

Author

Stanislav Sykora, Feb 15 2015

Keywords

• nonn
• base
• fini
• full

Comments

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

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

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.

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.

Examples

			6 is not a member because its expansion cannot be sliced in two.
The composite 73313 is a member because (7, 3313, 73, 313, 733, 13, 7331, 3) are all primes.
		

Crossrefs

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

Programs

Mathematica

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 *)

PARI

isComposite(n) = (n>2)&&(!isprime(n));
slicesIntoPrimes(n,b=10) = {my(k=b);if(n0,if(!isprime(n\k)||!isprime(n%k),return(0););k*=b;);1;}
isCompositeSlicingIntoPrimes(n,b=10) = isComposite(n) && slicesIntoPrimes(n,b);