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A085823 Numbers in which all substrings are primes. 67
2, 3, 5, 7, 23, 37, 53, 73, 373 (list; graph; refs; listen; history; text; internal format)



The definition implies that the number itself must be prime.

Apparently there are no such primes > 373.

From Jean-Marc Falcoz, Jan 11 2009: (Start)

This is correct.

There can't be any more terms, because they must necessarily be of the form

23737373733737... but the substring 237 is composite

or 273737373... but 273 is composite

or 5373737373... but 537 is composite

or 5737373737... but 573 is composite

or 37373737373... but 3737 is composite

or 7373737373... but 737 is composite

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.

Obviously, the digits 0, 1, 4, 6, 8, 9 can't appear anywhere in a term of the sequence. (End)

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.

From Hieronymus Fischer, Apr 20 2012: (Start)

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.

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)


Table of n, a(n) for n=1..9.

Onno M. Cain, Prime-bounded subwords, arXiv:1912.08598 [math.HO], 2019.

NRICH, Strange Numbers

Henri Picciotto's Math Education Page, "Super-slimes" in Infinity, Unit 1


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.


Select[Prime@ Range[10^3], AllTrue[FromDigits /@ Rest@ Subsequences@ IntegerDigits@ #, PrimeQ] &] (* Michael De Vlieger, Jul 30 2018 *)


Cf. A085822, A166504, A213300.

Sequence in context: A124674 A177061 A020994 * A284060 A211682 A100552

Adjacent sequences:  A085820 A085821 A085822 * A085824 A085825 A085826




Zak Seidov, Jul 04 2003


Thanks to Mark Underwood for pointing out misprints in the first version of this sequence.

Edited by N. J. A. Sloane, Jun 20 2009 at the suggestion of Lekraj Beedassy



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Last modified November 28 06:42 EST 2021. Contains 349401 sequences. (Running on oeis4.)