

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.


16



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

"73 is the largest integer with the property that all permutations of all of its substrings are primes."  M. Keith
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. 318321, Pour la Science, Paris 2000.


LINKS



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^(n1)); forvec(v=vector(n(n>2), i, [0, if(n>6, 9*(i+1)\n, n>3, 10(ni)\.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



KEYWORD

base,nonn


AUTHOR



EXTENSIONS



STATUS

approved



