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<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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 22 2009
EXTENSIONS
Corrected and extended by Chris K. Caldwell and R. J. Mathar, Mar 26 2009
STATUS
approved