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 A254751 Numbers such that, in base 10, all their proper prefixes and suffixes represent primes. 7
 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 237, 297, 313, 317, 373, 537, 597, 713, 717, 737, 797, 2337, 2397, 2937, 3113, 3137, 3173, 3797, 5937, 5997, 7197, 7337, 7397, 29397, 31373, 37937, 59397, 73313, 739397 (list; graph; refs; listen; history; text; internal format)
 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(n) in any way into two nonempty parts, each part represents a prime number. Every proper prefix of each member a(n) is a member of A024770, and every proper suffix is a member of A024785. Since these are finite sequences, a(n) is also finite. It has 45 members, the largest of which is 739397 and happens to be a prime. The sequence is a union of A254753 and A020994. A subsequence of A260181. - M. F. Hasler, Sep 16 2016 LINKS EXAMPLE 6 is not a member because its expansion cannot be sliced in two. 597 is a member because (5,97,59, and 7) are all primes. 2331 is excluded because 233 is prime, but 1 is not. - Gordon Hamilton, Feb 20 2015 MATHEMATICA fQ[n_] := (p = {2, 3, 5, 7}; If[ Union@ Join[p, {Mod[n, 10]}] != p, {False}, Block[{idn = IntegerDigits@ n, lng = Floor@ Log10@ n}, Union@ PrimeQ@ Flatten@ Table[{FromDigits[ Take[idn, i]], FromDigits[ Take[idn, -lng + i - 1]]}, {i, lng}] == {True}]]); Select[ Range@1000000, fQ] (* Robert G. Wilson v, Feb 21 2015 *) PROG (PARI) slicesIntoPrimes(n, b=10) = {my(k=b); if(n0, if(!isprime(n\k)||!isprime(n%k), return(0); ); k*=b; ); return(1); } (Sage) def breakIntoPrimes(n): ....D=n.digits() ....for i in [1..len(D)-1]: ........if not(is_prime(sum(D[i:][j]*10^j for j in range(len(D[i:])))) and is_prime(sum(D[:i][j]*10^j for j in range(len(D[:i]))))): ............return false ........else: ............continue ....return true R=[n for n in [10..1000000] if breakIntoPrimes(n)] # Tom Edgar, Feb 20 2015 CROSSREFS Cf. A020994, A024770, A024785, A254750, A254752, A254753, A254754, A254756. Cf. A260181. Sequence in context: A106582 A092619 A092624 * A260993 A276182 A091404 Adjacent sequences:  A254748 A254749 A254750 * A254752 A254753 A254754 KEYWORD nonn,base,fini,full AUTHOR Stanislav Sykora, Feb 15 2015 STATUS approved

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