A211686 Prime numbers > 10000 such that all the substrings of length >= 4 are primes (substrings with leading '0' are considered to be nonprime).

11093, 11171, 11933, 12011, 12239, 12377, 12791, 12917, 13037, 13217, 13613, 14519, 14591, 14813, 14831, 15233, 15791, 16073, 16091, 16217, 16673, 16691, 17333, 17417, 17477, 18233, 18311, 18713, 18719, 18731, 19013, 19319, 19739, 19973, 21319

Offset: 1

Author

Hieronymus Fischer, Jun 08 2012

Keywords

• nonn
• fini
• base

Comments

Only numbers > 10000 are considered, since all 4-digit primes are trivial members.

By definition, each term of the sequence with more than 5 digits is built up by an overlapped union of previous terms, i.e., a(254)=182339 has the two embedded previous terms a(26)=18233 and a(208)=82339.

The sequence is finite, the last term is 934919 (n=263). Proof of finiteness: Let p be a number with more than 6 digits. By the argument above, each 6-digit substring must be a previous term. The only 6-digit terms are 182339, 349199, 432713, 487793, 511933, 654799, 782339, 787793, 917333, 934919 (n=254..263, see b-file). As can be directly verified, none of them can be extended to a 7-digit number with the desired property.

Examples

			a(1)=11093, since all substrings of length >= 4 are primes (1109, 1093, and 11093).
a(263)=934919, all substrings of length >= 4 (9349, 3491, 4919, 93491, 34919 and 934919) are primes.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..263

Crossrefs

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Programs

Mathematica

sspQ[n_]:=Module[{idn=IntegerDigits[n],s1,s2},s1=FromDigits[Most[idn]];s2=FromDigits[Rest[idn]];IntegerLength[s1]==IntegerLength[s2]==4 && AllTrue[{s1,s2},PrimeQ]]; Select[Prime[Range[1230,9592]],sspQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* The program generates all 253 five-digit terms of the sequence *)(* Harvey P. Dale, May 11 2018 *)