A076730 Maximum number of (distinct) primes that an n-digit number may shelter (i.e., primes contained among all digital substrings' permutations).

1, 4, 11, 31, 106, 402, 1953, 10542, 64905, 362451, 2970505

Offset: 1

Author

Lekraj Beedassy, Nov 08 2002

Keywords

• hard
• more
• base
• nonn

Comments

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

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

Examples

			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).
		

Links

• M. Keith, Integers containing many embedded primes
• W. Schneider, MATHEWS, Primeval Numbers

Crossrefs

Cf. A072857, A076449, A076497, A134596 (largest n-digit primeval number).

Cf. A075053 (a variant of A039993), A134597 (= max A075053(1..10^n-1)).

Programs

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

Python

# see linked program in A076449

Formula

a(n) = A039993(A134596(n)) = max { A039993(m); m in A072857 and m < 10^n }. - M. F. Hasler, Mar 12 2014

a(n) = A076497(k) for k such that A072857(k) = A134596(n). - M. F. Hasler, Oct 15 2019

Extensions

Link fixed by Charles R Greathouse IV, Aug 13 2009

a(6) from M. F. Hasler, Mar 09 2014

a(7)-a(11) from Robert G. Wilson v, Mar 11 2014

a(9)-a(11) corrected by M. F. Hasler, Oct 15 2019