A072857 Primeval numbers: numbers that set a record for the number of distinct primes that can be obtained by permuting some subset of their digits.

1, 2, 13, 37, 107, 113, 137, 1013, 1037, 1079, 1237, 1367, 1379, 10079, 10123, 10136, 10139, 10237, 10279, 10367, 10379, 12379, 13679, 100279, 100379, 101237, 102347, 102379, 103679, 123479, 1001237, 1002347, 1002379, 1003679, 1012349, 1012379, 1023457, 1023467, 1023479, 1234579, 1234679, 10012349

Offset

1,2

Comments

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

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

Smallest monotonic increasing subsequence of A076449. - Lekraj Beedassy, Sep 23 2006

From M. F. Hasler, Oct 15 2019:

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.

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?

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

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

(End)

References

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

Links

Giovanni Resta, Table of n, a(n) for n = 1..100

C. K. Caldwell, The Prime Glossary, primeval number

J. P. Delahaye, Primes Hunters, 1379 is very primeval (in French)

M. Keith, Integers containing many embedded primes

W. Schneider, Primeval Numbers

N. J. A. Sloane, Transforms

G. Villemin's Almanach of Numbers, Nombre Primeval de Mike Keith

Wikipedia, Primeval number

Example

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.

Mathematica

(*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 *)

Prog

(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
(Python) # see linked program in A076449

Crossrefs

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

A076449 gives a similar sequence.

Cf. A119535 (prime subsequence).

Sequence in context: A291205 A005113 A239196 * A119535 A210849 A216155

Adjacent sequences: A072854 A072855 A072856 * A072858 A072859 A072860

Keyword

base,nonn

Author

Lekraj Beedassy, Jul 26 2002

Extensions

Edited, corrected and extended by Robert G. Wilson v, Nov 12 2002

Comment corrected by N. J. A. Sloane, Jan 25 2008

Status

approved