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 A158594 Numbers which yield a prime whenever a 3 is prefixed, appended or inserted. 11
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. REFERENCES 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 LINKS Jinyuan Wang, Table of n, a(n) for n = 1..500 EXAMPLE 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. MAPLE 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 PROG (PARI) isok(n)={i=#digits(n); m=1; k=0; while(k

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Last modified February 5 08:15 EST 2023. Contains 360082 sequences. (Running on oeis4.)