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%I #63 Oct 25 2023 10:14:07
%S 2,3,5,7,23,37,53,73,373
%N Numbers in which all substrings are primes.
%C The definition implies that the number itself must be prime.
%C Apparently there are no such primes > 373.
%C From _Jean-Marc Falcoz_, Jan 11 2009: (Start)
%C This is correct.
%C There can't be any more terms, because they must necessarily be of the form
%C 23737373733737... but the substring 237 is composite
%C or 273737373... but 273 is composite
%C or 5373737373... but 537 is composite
%C or 5737373737... but 573 is composite
%C or 37373737373... but 3737 is composite
%C or 7373737373... but 737 is composite
%C No other form is possible, otherwise, if the digit 2 or 5 is anywhere inside or at the end of the number, one substring-number is even or divisible by 5, and furthermore, there can't be twin digits, because one substring-number would then be divisible by 11.
%C Obviously, the digits 0, 1, 4, 6, 8, 9 can't appear anywhere in a term of the sequence. (End)
%C Subsequence of A024770 (right-truncatable primes), A068669 (noncomposite numbers in which all substrings are noncomposite). Supersequence of A202263 (primes in which all substrings and reversal substrings are primes). - _Jaroslav Krizek_, Jan 28 2012.
%C From _Hieronymus Fischer_, Apr 20 2012: (Start)
%C A more general definition is "Numbers such that all substrings of length <= 3 are primes". Proof: For numbers < 1000 this is plainly true. Suppose that there are such n >= 1000. We recognize that n must contain the string 373, as this is the only valid prime substring with the length 3. It follows, that there are substrings x37 or 73x, with any digit x. Evidently, neither x37 nor 73x are valid prime substrings, independent from the digit x. Thus, there is no number >= 1000 such that all substrings of length <= 3 are primes.
%C Also, numbers such that all substrings of length <= 2 are primes and the number of prime substrings of length = 3 is greater than m-3 for n <= 37373 and is greater than min(m-4,floor((m-1)/2) else; where m=floor(log_10(a(n)))+1 = number of digits. (End)
%H Onno M. Cain, <a href="https://arxiv.org/abs/1912.08598">Prime-bounded subwords</a>, arXiv:1912.08598 [math.HO], 2019.
%H NRICH, <a href="http://nrich.maths.org/722">Strange Numbers</a>
%H Henri Picciotto's Math Education Page, <a href="http://www.mathedpage.org/infinity/infinity-1.pdf">"Super-slimes" in Infinity, Unit 1</a>
%e Example : 373 is in the sequence, because 3, 7, 37, 73 and 373 are prime, but 733 is not in the sequence, because 33 is not prime.
%t Select[Prime@ Range[10^3], AllTrue[FromDigits /@ Rest@ Subsequences@ IntegerDigits@ #, PrimeQ] &] (* _Michael De Vlieger_, Jul 30 2018 *)
%Y Cf. A085822, A166504, A213300.
%K nonn,fini,full,base
%O 1,1
%A _Zak Seidov_, Jul 04 2003
%E Thanks to _Mark Underwood_ for pointing out misprints in the first version of this sequence.
%E Edited by _N. J. A. Sloane_, Jun 20 2009 at the suggestion of _Lekraj Beedassy_
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