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A158594
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Numbers which yield a prime whenever a 3 is prefixed, appended or inserted.
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11
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1, 7, 11, 17, 31, 37, 73, 121, 271, 331, 343, 359, 361, 373, 533, 637, 673, 733, 793, 889, 943, 1033, 1183, 2297, 3013, 3119, 3223, 3353, 3403, 3461, 3757, 3827, 3893, 3923, 4313, 4543, 4963, 5323, 5381, 5419, 6073, 6353, 8653, 9103, 9887, 10423, 14257
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OFFSET
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1,2
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COMMENTS
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1) It is conjectured that sequences of this type are infinite; also that an infinite number of primes is included.
2) Necessarily a(n) has end digit 1,3,7 or 9.
3) Sum of digits of a(n) has form 3k-1 or 3k+1.
4) Sequence is part of A068674 a(n) n=1,...,30: first 14 primes: 7, 11, 17, 31, 37, 73, 271, 331, 359, 373, 673, 733, 2297, 3461.
5) Note the "world record" 2297: smallest prime which yields five other primes 32297, 23297, 22397, 22937, 22973.
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REFERENCES
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Marcus Du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins. 2004
Bryan Bunch, Kingdom of Infinite Number: A Field Guide, W.H. Freeman & Company, 2001
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LINKS
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EXAMPLE
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109 is not a term: 3109, 1039, 1093 are primes, but 1309 = 7 * 11 * 17.
121 is a term: 3121 (3 prefixed), 1213 (3 appended), 1321 and 1231 (3 inserted) are primes.
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MAPLE
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Lton := proc(L) local i ; add(op(i, L)*10^(i-1), i=1..nops(L) ) ; end: isA158594 := proc(n) local dgs, i, p; dgs := convert(n, base, 10) ; p := [3, op(dgs)] ; if not isprime(Lton(p)) then RETURN(false) ; fi; p := [op(dgs), 3] ; if not isprime(Lton(p)) then RETURN(false) ; fi; for i from 1 to nops(dgs)-1 do p := [op(1..i, dgs), 3, op(i+1..nops(dgs), dgs)] ; if not isprime(Lton(p)) then RETURN(false) ; fi; od: RETURN(true) ; end: for n from 1 to 25000 do if isA158594(n) then printf("%d, ", n) ; fi; od: # R. J. Mathar, Mar 26 2009
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PROG
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(PARI) isok(n)={i=#digits(n); m=1; k=0; while(k<i+1&&m==1, r=n\10^k; s=n-r*10^k; t=r*10^(k+1)+s+3*10^k; if(isprime(t)==0, m=0); k++); m; } \\ Jinyuan Wang, Feb 02 2019
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CROSSREFS
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Cf. A068674, Numbers which yield primes when a 3 is prefixed or appended.
Cf. A068679, Numbers which yield a prime whenever a 1 is inserted anywhere in them (including at the beginning or end).
Cf. A158232, Numbers which yield primes when "13" is prefixed or appended.
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KEYWORD
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nonn,base
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AUTHOR
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Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 22 2009
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EXTENSIONS
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STATUS
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approved
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