A056709 Naught-y primes, primes with noughts (or zeros).

101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1201, 1301, 1303, 1307, 1409, 1601, 1607, 1609, 1709, 1801, 1901, 1907

Offset: 1

Author

Robert G. Wilson v, Aug 10 2000

Keywords

• nonn
• base

Comments

Intersection of A000040 and A011540. - Michel Marcus, Mar 12 2015

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Chris Caldwell, Naughty prime, Prime Pages' Glossary (UTM). (Date?)

Crossrefs

Cf. A000040 (primes), A011540 (numbers that contain a 0).

Complement, in A000040, of zeroless primes A038618.

Cf. A164968 (Naughty primes: most digits are 0).

Programs

Magma

[p:p in PrimesUpTo(2000)|0 in Intseq(p)]; // Marius A. Burtea, Jan 13 2020

Mathematica

Select[ Range[ 1, 2500, 2 ], PrimeQ[ # ] && Sort[ RealDigits[ # ][ [ 1 ] ] ][ [ 1 ] ] == 0 & ]
(* Second program: *)
Select[Prime@ Range@ 300, DigitCount[#, 10, 0] > 0 &] (* Michael De Vlieger, Jan 28 2020 *)

PARI

is(n)=isprime(n)&&vecsort(eval(Vec(Str(n))),,8)[1]==0

PARI

select( {is_A056709(n)=!vecmin(digits(n))&&isprime(n)}, [1..2000]) \\ Defines the characteristic function is_A; as check & example: select terms in [1..2000].
next_A056709(n)={until(!vecmin(digits(n)), n=nextprime(n+1));n} \\ Successor function: find smallest a(k) > n. Useful to get a vector of consecutive terms:
A056709_vec(n,M=99)=M--;vector(n,i,M=next_A056709(M)) \\ get n terms >= M (if given, else start with a(1)).  \\ M. F. Hasler, Jan 12 2020

Python

from sympy import primerange
def aupto(lim): return [p for p in primerange(1, lim+1) if '0' in str(p)]
print(aupto(1910)) # Michael S. Branicky, Mar 11 2022

Formula

a(n) ~ n log n: almost all primes are in this sequence. - Charles R Greathouse IV, Jul 24 2012