|
%I #43 Jun 16 2021 17:40:57
%S 1,4,11,31,106,402,1953,10542,64905,362451,2970505
%N Maximum number of (distinct) primes that an n-digit number may shelter (i.e., primes contained among all digital substrings' permutations).
%C See sequence A134596 for the least numbers of given length which yields these maxima over n-digit indices for A039993. - _M. F. Hasler_, Mar 11 2014
%C By definition this is a subsequence of A076497. The term a(10) was incorrectly given as 398100 = A075053(1123456789), which double-counts each prime using only one digit '1'. But a(10) = A039993(1123456789) = A076497(80) = 362451. The values given for a(9) and a(11) were also incorrect, the latter probably for the same reason, and for a(9) probably due to double-counting of primes with leading zeros. - _M. F. Hasler_ and _David A. Corneth_, Oct 15 2019
%H M. Keith, <a href="http://www.cadaeic.net/primeval.htm">Integers containing many embedded primes</a>
%H W. Schneider, MATHEWS, <a href="http://web.archive.org/web/2004/www.wschnei.de/digit-related-numbers/primeval-numbers.html">Primeval Numbers</a>
%F a(n) = A039993(A134596(n)) = max { A039993(m); m in A072857 and m < 10^n }. - _M. F. Hasler_, Mar 12 2014
%F a(n) = A076497(k) for k such that A072857(k) = A134596(n). - _M. F. Hasler_, Oct 15 2019
%e We have a(3)=11, since among numbers 100 through 999, the smallest ones having 5, 6, 7, 8, 10, 11 embedded primes are respectively 107, 127, 113, 167, 179, 137 (the last of these being the first reaching the maximum number of 11 embedded primes, viz. 3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 317).
%o (PARI) a(n,m=0)=for(k=10^(n-1),10^n-1,A039993(k)>m&&m=A039993(k));m \\ _M. F. Hasler_, Mar 09 2014
%o (Python) # see linked program in A076449
%Y Cf. A072857, A076449, A076497, A134596 (largest n-digit primeval number).
%Y Cf. A075053 (a variant of A039993), A134597 (= max A075053(1..10^n-1)).
%K hard,more,base,nonn
%O 1,2
%A _Lekraj Beedassy_, Nov 08 2002
%E Link fixed by _Charles R Greathouse IV_, Aug 13 2009
%E a(6) from _M. F. Hasler_, Mar 09 2014
%E a(7)-a(11) from _Robert G. Wilson v_, Mar 11 2014
%E a(9)-a(11) corrected by _M. F. Hasler_, Oct 15 2019
|
