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A254751
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Numbers such that, in base 10, all their proper prefixes and suffixes represent primes.
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7
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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
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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: 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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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 *)
Select[Range[10, 750000], AllTrue[Flatten[Table[FromDigits/@TakeDrop[IntegerDigits[#], n], {n, IntegerLength[#]-1}]], PrimeQ]&] (* Harvey P. Dale, Feb 13 2024 *)
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PROG
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(PARI) slicesIntoPrimes(n, b=10) = {my(k=b); if(n<b, return(0); ); while(n\k>0, 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
[n for n in [10..1000] if breakIntoPrimes(n)] # Tom Edgar, Feb 20 2015
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CROSSREFS
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KEYWORD
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nonn,base,fini,full
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AUTHOR
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STATUS
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approved
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