A134809 -id:A134809 - cp's OEIS frontend

A309488 Primes whose decimal expansion is of the form d_1+0+d_2+0+d_3+0+...+0+d_k where d_i are digits with 1 <= d_i <= 9, k > 1 and + stands for concatenation.

101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 10103, 10301, 10303, 10501, 10601, 10607, 10709, 10903, 10909, 20101, 20107, 20201, 20407, 20507, 20509, 20707, 20807, 20809, 20903, 30103, 30109, 30203, 30307, 30403, 30509, 30703, 30707, 30803, 30809

Offset: 1

Bernard Schott, Aug 04 2019

The terms of this sequence have necessarily an odd number >= 3 of digits.
There is only one term whose digits > 0 are all equal: 101.
The only cyclops primes (A134809) of this sequence are the first 15 terms from 101 to 907.
The first palindromes of this sequence are 101, 10301, 10501, 10601, 30103, 30203, 30403, 30703, 30803, ...
Intersection with A309101 = {503, 10103, 10303, 10903, ...}.

			103 is the smallest term with two distinct digits > 0.
10607 is the smallest term with three distinct digits > 0.

Cf. A000040, A002385, A134809, A309101.

Subsequence of A059168 (undulating primes).

sol:=[]; m:=1; for p in PrimesInInterval(101,50000) do  v:=Reverse(Intseq(p)); test:=0; for u in [1..#v-1] do if u mod 2 eq 0 and v[u] eq 0 and v[u+1] ne 0 then test:=test+1; end if; end for; if test eq (#v-1)/2 then sol[m]:=p; m:=m+1; end if; end for; sol; // Marius A. Burtea, Aug 04 2019

aQ[n_] := PrimeQ[n] && OddQ[(m = Length[(d = IntegerDigits[n])])] && Flatten@Position[d, ?(# == 0 &)] == Range[2, m, 2]; Select[Range[100, 31000], aQ] (* _Amiram Eldar, Aug 04 2019 *)

eva(n) = subst(Pol(n), x, 10)
f(n) = my(d=digits(n)); eva(vector(2*#d-1, k, if (k%2, d[1+k\2]))) \\ from Michel Marcus
terms(n) = my(i=0); for(k=10, oo, if(i>=n, break); if(vecmin(digits(k)) > 0, my(iz=f(k)); if(ispseudoprime(iz), print1(iz, ", "); i++)))
/* Print initial 40 terms as follows: */
terms(40) \\ Felix Fröhlich, Aug 08 2019

A345728 Primes with an odd number of digits and 0 as the middle digit.

Original entry on oeis.org

101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 10007, 10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, 11003, 11027, 11047, 11057, 11059, 11069, 11071, 11083, 11087, 11093, 12007, 12011, 12037, 12041, 12043, 12049, 12071, 12073, 12097, 13001, 13003, 13007

Offset: 1

James S. DeArmon, Jun 27 2021

Cf. A000040 (primes), A134809 (Cyclops primes).

Select[Prime@Range@2000,OddQ[d=Length[s=IntegerDigits[#]]]&&s[[Ceiling[d/2]]]==0&] (* Giorgos Kalogeropoulos, Jul 04 2021 *)

isok(p) = {if (isprime(p), my(d=digits(p)); (#d % 2) && (d[#d\2+1] == 0););} \\ Michel Marcus, Jun 28 2021

#!/usr/bin/perl
$str = "";
foreach $cand (101..20000){  # loop over candidates
    next unless &isPrime($cand);  # is $cand prime? 0/1 result
    @a = split("",$cand);
    next if @a/2 == int @a/2;
    $mid = int @a/2;
    next unless $a[$mid] == 0;
    $str .= "$cand, ";
}
chop $str; chop $str;
print "$str\n";

from sympy import isprime
from itertools import product
def agen(maxdigits):
    for digits in range(3, maxdigits+1, 2):
        for first in "123456789":
            for left in product("0123456789", repeat=digits//2-1):
                left = "".join(left)
                for right in product("0123456789", repeat=digits//2-1):
                    right = "".join(right)
                    for last in "1379":
                        t = int("".join(first + left + "0" + right + last))
                        if isprime(t): yield t
print([an for an in agen(5)]) # Michael S. Branicky, Jun 28 2021

cp's OEIS Frontend

A309488 Primes whose decimal expansion is of the form d_1+0+d_2+0+d_3+0+...+0+d_k where d_i are digits with 1 <= d_i <= 9, k > 1 and + stands for concatenation.

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