

A254753


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


7



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
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OFFSET

1,1


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.


LINKS

Table of n, a(n) for n=1..34.


EXAMPLE

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.


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 *)


PROG

(PARI) isComposite(n) = (n>2)&&(!isprime(n));
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; }
isCompositeSlicingIntoPrimes(n, b=10) = isComposite(n) && slicesIntoPrimes(n, b);


CROSSREFS

Cf. A020994, A024770, A024785, A254750, A254751, A254752, A254754.
Sequence in context: A121609 A092631 A061371 * A070809 A280646 A295799
Adjacent sequences: A254750 A254751 A254752 * A254754 A254755 A254756


KEYWORD

nonn,base,fini,full


AUTHOR

Stanislav Sykora, Feb 15 2015


STATUS

approved



