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A254754
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Prime numbers such that, in base 10, all their proper prefixes and suffixes represent composites.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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PROG
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(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);
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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