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

1,1

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).

Links

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A244777 A107192 A061372 * A083473 A142186 A142757

Adjacent sequences: A254751 A254752 A254753 * A254755 A254756 A254757

Keyword

nonn,base

Author

Stanislav Sykora, Feb 15 2015

Status

approved