A158594 Numbers which yield a prime whenever a 3 is prefixed, appended or inserted.

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

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 22 2009

Keywords

• nonn
• base

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.

Examples

			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.
		

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

Crossrefs

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.

Programs

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

PARI

isok(n)={i=#digits(n);m=1;k=0;while(kJinyuan Wang, Feb 02 2019

Extensions

Corrected and extended by Chris K. Caldwell and R. J. Mathar, Mar 26 2009