A072857 - OEIS

%I #51 Nov 16 2024 11:41:25

%S 1,2,13,37,107,113,137,1013,1037,1079,1237,1367,1379,10079,10123,

%T 10136,10139,10237,10279,10367,10379,12379,13679,100279,100379,101237,

%U 102347,102379,103679,123479,1001237,1002347,1002379,1003679,1012349,1012379,1023457,1023467,1023479,1234579,1234679,10012349

%N Primeval numbers: numbers that set a record for the number of distinct primes that can be obtained by permuting some subset of their digits.

%C RECORDS transform of A039993. - _N. J. A. Sloane_, Jan 25 2008. See A239196 and A239197 for the RECORDS transform of the closely related sequence A075053. - _M. F. Hasler_, Mar 12 2014

%C "73 is the largest integer with the property that all permutations of all of its substrings are primes." - M. Keith

%C Smallest monotonic increasing subsequence of A076449. - _Lekraj Beedassy_, Sep 23 2006

%C From _M. F. Hasler_, Oct 15 2019:

%C All terms > 37 start with leading digit 1 and have all other digits in nondecreasing order. The terms are smallest representatives of the class of numbers having the same digits, cf. A179239 and A328447 which both contain this as a subsequence.

%C The frequency of primes is roughly 50% for the displayed values, but appears to decrease. Can it be proved that the asymptotic density is zero?

%C Can we prove that there are infinitely many even terms? (Of the form 10...01..12345678?)

%C Can it be proved that there is no term that is a multiple of 3? (Or the contrary? Are there infinitely many?)

%C (End)

%D J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), "1379's quite primeval, is it not?", pp. 318-321, Pour la Science, Paris 2000.

%H Giovanni Resta, <a href="/A072857/b072857.txt">Table of n, a(n) for n = 1..100</a>

%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=Primeval">primeval number</a>

%H J. P. Delahaye, Primes Hunters, <a href="http://web.archive.org/web/20020901024608/http://www.pour-la-science.com/numeros/pls-258/logique.htm">1379 is very primeval (in French)</a>

%H M. Keith, <a href="http://www.cadaeic.net/primeval.htm">Integers containing many embedded primes</a>

%H W. Schneider, <a href="http://web.archive.org/web/2004/www.wschnei.de/digit-related-numbers/primeval-numbers.html">Primeval Numbers</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H G. Villemin's Almanach of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Premier/primeval.htm">Nombre Primeval de Mike Keith</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Primeval_number">Primeval number</a>

%e 1379 is in the sequence because it is the smallest number whose digital permutations form a total of 31 primes, viz. 3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97, 137, 139, 173, 179, 193, 197, 317, 379, 397, 719, 739, 937, 971, 1973, 3719, 3917, 7193, 9137, 9173, 9371.

%t (*first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Length[ Select[ FromDigits /@ Flatten[ Permutations /@ Subsets[ IntegerDigits[ n]], 1], PrimeQ[ # ] &]]; d = -1; Do[ b = f[n]; If[b > d, Print[n]; d = b], {n, 2^20}] (* _Robert G. Wilson v_, Feb 12 2005 *)

%t Join[{1},DeleteDuplicates[Table[{n,Count[Union[FromDigits/@Flatten[Permutations[#]&/@Subsets[IntegerDigits[n]],1]],_?PrimeQ]},{n,2,125000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]]] (* The program generates the first 30 terms of the sequence. *) (* _Harvey P. Dale_, Nov 16 2024 *)

%o (PARI) A072857_upto(num_digits,s=1,m=-1,L=List())={for(n=s,num_digits, my(u=10^(n-1)); forvec(v=vector(n-(n>2),i,[0,if(n>6,9*(i+1)\n,n>3,10-(n-i)\.6,7)]), m<A039993(u+fromdigits(v)) && m=A039993(listput(L,u+fromdigits(v))),1)); Vec(L)} \\ Optional 2nd and 3rd arg allow to extend a previous computation. - _M. F. Hasler_, Oct 15 2019

%o (Python) # see linked program in A076449

%Y Cf. A039993, A075053, A076497, A239196, A239197.

%Y A076449 gives a similar sequence.

%Y Cf. A119535 (prime subsequence).

%K base,nonn

%O 1,2

%A _Lekraj Beedassy_, Jul 26 2002

%E Edited, corrected and extended by _Robert G. Wilson v_, Nov 12 2002

%E Comment corrected by _N. J. A. Sloane_, Jan 25 2008