A254754 Prime numbers such that, in base 10, all their proper prefixes and suffixes represent composites.

89, 409, 449, 499, 809, 4049, 4549, 4649, 4909, 4969, 6299, 6469, 6869, 6899, 6949, 8009, 8039, 8069, 8209, 8609, 8669, 8699, 8849, 9049, 9209, 9649, 9949, 40009, 40099, 40609, 40639, 40699, 40849, 42209, 42649, 44249, 44699, 45949, 46049, 46099

Offset: 1

Author

Stanislav Sykora, Feb 15 2015

Keywords

• nonn
• base

Comments

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

Alternative definition: Slice the decimal expansion of the prime number a(n) in any way into two nonempty parts; then both parts represent a composite number.

This sequence is a subset of A254750. Each member a(n) must start with one of the digits {4,6,8,9} and end with 9.

Every proper prefix of each member a(n) is a member of A202260, and every proper suffix is a member of A254755.

These numbers are rare and tend to become rarer with increasing n, but the sequence does not seem to terminate (for example, 4*10^28 + 9 is a member).

Examples

			7 is not a member because its expansion cannot be sliced in two.
The prime 4969 is a member because it is a prime and the slices (4, 969, 49, 69, 496, and 9) are all composites.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A202260, A254750, A254751, A254752, A254753, A254755.

Programs

Mathematica

Select[Prime[Range[5,5000]],AllTrue[Flatten[Table[FromDigits/@TakeDrop[IntegerDigits[#],n],{n,IntegerLength[ #]-1}]],CompositeQ]&] (* Harvey P. Dale, Sep 22 2024 *)

PARI

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