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

1,1

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

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.

Example

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

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 *)

Prog

(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], i<z, [2, n-1], [n, n]))); forvec(d=vector(z, i, [1, 9]), bittest(650, d[z])&& vecsum(d)%3&& forvec(p=v, isprime(d*vecextract(t, p))&& listput(a, d*vecextract(t, p)), 2))); Set(a)} \\ M. 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

Crossrefs

Sequence in context: A330135 A031598 A210760 * A165296 A182697 A101442

Adjacent sequences: A164965 A164966 A164967 * A164969 A164970 A164971

Keyword

base,nonn

Author

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

Status

approved