A164968 Naughty primes: primes in which the number of zeros is greater than the number of all other digits.

10007, 10009, 40009, 70001, 70003, 70009, 90001, 90007, 100003, 200003, 200009, 300007, 400009, 500009, 700001, 900001, 900007, 1000003, 1000033, 1000037, 1000039, 1000081, 1000099, 1000303, 1000403, 1000409, 1000507, 1000609, 1000907, 1001003, 1003001

Offset: 1

Author

G. L. Honaker, Jr., Sep 02 2009

Keywords

• base
• nonn

Comments

a(31) = 1003001 is the smallest palindromic naughty prime. - M. F. Hasler, Nov 22 2009

This sequence can be considered as irregular table in which row n lists the terms with n digits. The row lengths (number of terms with n digits) are then 0, 0, 0, 0, 8, 9, 296, 275, 7934, 9527, 235729, ... - M. F. Hasler, Jul 13 2018

Examples

			a(24) = 1000303 is a naughty prime because the number of zeros is greater than the number of all other digits.
		

Links

• M. F. Hasler and Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000 (first 5000 terms from M. F. Hasler).
• Chris Caldwell, The Prime Glossary, Naughty prime.

Programs

Maple

Q[1]:= [seq([i],i=1..9)]:
for d from 2 to 6 do Q[d]:= map(t -> seq([i,op(t)],i=1..9),Q[d-1]) od:
F:= proc(d) local R,dn,s,sp,q,x;
   R:= NULL;
   for dn from 2 to floor((d-1)/2) do
      for s in combinat:-choose([$1..d-2],dn-2) do
        sp:= [0,op(s),d-1];
        for q in Q[dn] do
          x:= add(q[i]*10^sp[i],i=1..dn);
          if isprime(x) then R:= R, x fi;
    od od od;
    sort([R])
end proc:
seq(op(F(d)),d=5..7); # Robert Israel, Jul 10 2018

Mathematica

lst = {}; Do[If[PrimeQ[n] && Count[IntegerDigits[n], 0] > IntegerLength[n]/2, AppendTo[lst, n]], {n, 10^4 + 1, 3^13, 2}]; lst (* Arkadiusz Wesolowski, Sep 18 2011 *)
Select[Prime[Range[100000]],DigitCount[#,10,0]>IntegerLength[#]/2&] (* Harvey P. Dale, Jun 09 2015 *)

PARI

next_A164968(p)={ for( n=#Str(p)\2+1,oo, my(L=10^(2*n+1)); p=max(10^(2*n-3),p); while( L>p=nextprime(p+1), vecsort(Vecsmall(Str(p)))[n]>48 || return(p));p=0) } \\ M. F. Hasler, Nov 22 2009, syntax update Jul 10 2018

PARI

A164968_row(n, a=List(), t=vectorv(n, i, 10^(n-i)))={for(z=2, (n-1)\2, my(v=vector(z, i, if(i<2, [1, 1], iM. F. Hasler, Jul 13 2018

Python

from sympy import isprime
from itertools import combinations, count, islice, product
def agen(): # generator of terms
    for d in count(5):
        for first in "123456789":
            passed = set()
            for last in "1379":
                for znum in range(d//2 + 1, d-1):
                    for zlocs in combinations(range(d-2), znum):
                        for rest in product("123456789", repeat=d-2-znum):
                            nzi, middle = 0, []
                            for i in range(d-2):
                                if i in zlocs:
                                    middle.append("0")
                                else:
                                    middle.append(rest[nzi])
                                    nzi += 1
                            t = int("".join(first + "".join(middle) + last))
                            if isprime(t):
                                passed.add(t)
            yield from sorted(passed)
print(list(islice(agen(), 31))) # Michael S. Branicky, Mar 11 2022