cp's OEIS Frontend

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

%I A254754 #13 Sep 22 2024 15:01:03
%S A254754 89,409,449,499,809,4049,4549,4649,4909,4969,6299,6469,6869,6899,6949,
%T A254754 8009,8039,8069,8209,8609,8669,8699,8849,9049,9209,9649,9949,40009,
%U A254754 40099,40609,40639,40699,40849,42209,42649,44249,44699,45949,46049,46099
%N A254754 Prime numbers such that, in base 10, all their proper prefixes and suffixes represent composites.
%C A254754 A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
%C A254754 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.
%C A254754 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.
%C A254754 Every proper prefix of each member a(n) is a member of A202260, and every proper suffix is a member of A254755.
%C A254754 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).
%H A254754 Stanislav Sykora, <a href="/A254754/b254754.txt">Table of n, a(n) for n = 1..20000</a>
%e A254754 7 is not a member because its expansion cannot be sliced in two.
%e A254754 The prime 4969 is a member because it is a prime and the slices (4, 969, 49, 69, 496, and 9) are all composites.
%t A254754 Select[Prime[Range[5,5000]],AllTrue[Flatten[Table[FromDigits/@TakeDrop[IntegerDigits[#],n],{n,IntegerLength[ #]-1}]],CompositeQ]&] (* _Harvey P. Dale_, Sep 22 2024 *)
%o A254754 (PARI) isComposite(n) = (n>2)&&(!isprime(n));
%o A254754 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);}
%o A254754 isPrimeSlicingIntoComposites(n,b=10) = isprime(n) && slicesIntoComposites(n,b);
%Y A254754 Cf. A202260, A254750, A254751, A254752, A254753, A254755.
%K A254754 nonn,base
%O A254754 1,1
%A A254754 _Stanislav Sykora_, Feb 15 2015