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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<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

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.

Sequence in context: A019418 A068674 A156112 * A215419 A107642 A079651

Adjacent sequences: A158591 A158592 A158593 * A158595 A158596 A158597

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

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