A056709 -id:A056709 - cp's OEIS frontend

A256186 Naught-y primes (A056709) that after removing all zeros become zeroless primes (A038618).

101, 103, 107, 109, 307, 401, 503, 509, 601, 607, 701, 709, 809, 907, 1009, 1013, 1031, 1039, 1049, 1051, 1063, 1091, 1093, 1097, 1103, 1301, 1307, 1409, 1607, 1709, 1801, 1901, 1907, 2003, 2011, 2027, 2029, 2039, 2063, 2069, 2081, 2083, 2203, 2207, 2309, 2609, 2707

Offset: 1

Zak Seidov, Mar 19 2015

Subsequence of A256227.

			a(1)=101=A056709(1) => 11=A038618(5), a(15)=1009=A056709(16) => 19=A038618(8).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A056709 Naught-y primes, primes with noughts (or zeros), A038618 Primes not containing digit '0', a.k.a. zeroless primes.

ss = {}; Do[id = IntegerDigits[p = Prime[k]]; If[Min[id] < 1 && PrimeQ[FromDigits[Delete[id, Position[id, 0]]]], ss = {ss, p}], {k, 1, 500}]; Flatten[ss] (* Zak Seidov *)
Select[Prime[Range[500]], DigitCount[#, 10, 0] > 0 && PrimeQ[FromDigits[DeleteCases[IntegerDigits[#], 0]]] &] (* Alonso del Arte, Mar 22 2015 *)

is(n)=my(d=digits(n)); isprime(n) && #d>#(d=select(x->x,d)) && isprime(fromdigits(d)) \\ Charles R Greathouse IV, Mar 19 2015

A373294 a(n) is the number of n-digit primes that have at least one zero among their digits (A056709).

Original entry on oeis.org

0, 0, 15, 204, 2251, 23715, 240528, 2391394, 23540109, 230318080, 2244729936, 21819401038, 211711461260, 2051836712085

Offset: 1

Gonzalo Martínez, May 30 2024

			For n = 3, the 3-digit prime numbers that have the digit 0 are 101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809 and 907. Therefore, a(3) = 15.

First differences of A091644.

Cf. A000040, A006879, A056709.

a(n) = my(s=0); forprime(p=10^(n-1), 10^n-1, if (vecmin(digits(p)) == 0, s++)); s; \\ Michel Marcus, May 31 2024

a(n) = A091644(n) - A091644(n-1) for n > 1. - Michael S. Branicky, May 31 2024

More terms (using A091644) from Michael S. Branicky, May 30 2024

A038618 Primes not containing the digit '0'.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Complement of A056709 with respect to primes (A000040). - Lekraj Beedassy, Jul 04 2010

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
René-Louis Clerc, Nombres premiers primaires et nombres premiers secondaires , 2025.
M. F. Hasler, Numbers avoiding certain digits, OEIS Wiki, Jan 12 2020.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).
Eric Weisstein's World of Mathematics, Zerofree.
Index to entries for primes with digits in a given set

Cf. A010051, A168046, A384793, A384794.
Subsequence of A000040 (primes), A052382 (zeroless numbers) and A195943.
Primes having no digit d = 0..9 are this sequence, A038603, A038604, A038611, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.

a038618 n = a038618_list !! (n-1)
a038618_list = filter ((== 1) . a168046) a000040_list
-- Reinhard Zumkeller, Apr 07 2014, Sep 27 2011

[ p: p in PrimesUpTo(300) | not 0 in Intseq(p) ];  // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 0] == 0 &] (* Vincenzo Librandi, Aug 09 2011 *)

is(n)=if(isprime(n),n=vecsort(eval(Vec(Str(n))),,8);n[1]>0) \\ Charles R Greathouse IV, Aug 09 2011

lista(nn) = forprime (p=2, nn, if (vecmin(digits(p)), print1(p, ", "))); \\ Michel Marcus, Apr 06 2016

next_A038618(n)=until(vecmin(digits(n=nextprime(next_A052382(n)))),);n \\ Cf. OEIS Wiki page (LINKS) for other programs. - M. F. Hasler, Jan 12 2020

from sympy import primerange
def aupto(N): return [p for p in primerange(1, N+1) if '0' not in str(p)]
print(aupto(300)) # Michael S. Branicky, Mar 11 2022

Intersection of A052382 (zeroless numbers) and A000040 (primes); A168046(a(n))*A010051(a(n)) = 1. - Reinhard Zumkeller, Dec 01 2009

a(n) ≍ n^(log 10/log 9) log n. - Charles R Greathouse IV, Aug 03 2023

A134809 Cyclops primes.

Original entry on oeis.org

101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 11027, 11047, 11057, 11059, 11069, 11071, 11083, 11087, 11093, 12011, 12037, 12041, 12043, 12049, 12071, 12073, 12097, 13033, 13037, 13043, 13049, 13063

Offset: 1

Omar E. Pol, Nov 25 2007

Cyclops numbers that are prime numbers: primes with an odd number of digits with middle digit 0 that have only one digit 0.
The only known Fibonacci number in this sequence is 99194853094755497 (see A005478 and A182809).
The only known Lucas number in this sequence is 688846502588399 (see A005479 and A182811).

Harvey P. Dale, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 688846502588399
G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 99194853094755497

Intersection of prime numbers A000040 and cyclops numbers A134808.

Cf. A005478, A005479, A056709, A069675, A182809, A182811.

(* First run the program given for A134808 *) Select[Prime[Range[2000]], cyclopsQ] (* Alonso del Arte, Dec 16 2010 *)
cycQ[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn];OddQ[len] && Count[idn,0] == 1 && idn[[(len+1)/2]]==0]; Select[Flatten[Table[Prime[ Range[ PrimePi[10^(2n)+1],PrimePi[10^(2n+1)]]],{n,2}]],cycQ] (* Harvey P. Dale, Jun 20 2014 *)

# cyclops() in A134808
from sympy import isprime
print([c for c in cyclops(upto=13063) if isprime(c)]) # Michael S. Branicky, Jan 05 2021

Links added by Omar E. Pol, Mar 25 2011

A157677 Primes p such that p + (product of digits of p) is also prime.

Original entry on oeis.org

23, 29, 61, 67, 83, 101, 103, 107, 109, 163, 233, 239, 283, 293, 307, 347, 349, 401, 409, 431, 439, 443, 449, 499, 503, 509, 563, 569, 601, 607, 613, 617, 619, 653, 659, 677, 683, 701, 709, 743, 809, 907, 929, 941, 1009, 1013, 1019, 1021, 1031, 1033, 1039

Offset: 1

Kyle D. Balliet, Mar 04 2009

If p contains a zero, then p is trivially a member.

			83 is prime, and 83 + 8*3 = 89 which is also prime. 103 is prime, and 103 + 1*0*3 = 103 is also prime. Thus 89 and 103 are members.

Robert Israel, Table of n, a(n) for n = 1..10000

Union of A092518 and A056709.

Cf. A225303.

a := proc (n) local nn: nn := convert(ithprime(n), base, 10): if isprime(ithprime(n)+product(nn[j], j = 1 .. nops(nn))) = true then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 180); # Emeric Deutsch, Mar 08 2009

Select[Prime[Range[175]], PrimeQ[# + Times @@ IntegerDigits[#]] &] (* Jayanta Basu, Apr 22 2013 *)

dprod(n)=n=digits(n); prod(i=1,#n,n[i])
is(n)=isprime(n) && isprime(n+dprod(n)) \\ Charles R Greathouse IV, Dec 27 2013

a(n) ~ n log n. - Charles R Greathouse IV, Apr 22 2013

More terms from Emeric Deutsch, Mar 08 2009

A101987 Product of nonzero digits of n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 1, 3, 7, 9, 6, 18, 3, 21, 4, 12, 28, 15, 45, 6, 42, 7, 21, 63, 24, 72, 63, 1, 3, 7, 9, 3, 14, 3, 21, 27, 36, 5, 35, 18, 42, 21, 63, 8, 9, 27, 63, 81, 2, 12, 28, 36, 18, 54, 8, 10, 70, 36, 108, 14, 98, 16, 48, 54, 21, 3, 9, 21, 9, 63, 84, 108, 45, 135, 126, 63, 189, 72, 216

Offset: 1

Zak Seidov, Jan 29 2005

First differs from A053666 in 26th term.

			a(25) = 63 because the 25th prime is 97 and 9 * 7 = 63.
a(26) = 1 because the 26th prime is 101, but we ignore the 0 and thus have 1 * 1 = 1.

Cf. A051801, A053666.

Cf. A038618, A056709.

a:= n-> mul(`if`(i=0, 1, i), i=convert(ithprime(n), base, 10)):
seq(a(n), n=1..77);  # Alois P. Heinz, Mar 11 2022

Table[Times@@ReplaceAll[IntegerDigits[Prime[n]], 0 -> 1], {n, 80}] (* Alonso del Arte, Feb 28 2014 *)

a(n) = vecprod(select(x->(x>1), digits(prime(n)))); \\ Michel Marcus, Mar 11 2022

from math import prod
from sympy import sieve
def A051801(n): return prod(int(d) for d in str(n) if d != '0')
def a(n): return A051801(sieve[n])
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Mar 11 2022

a(n) = A051801(prime(n)). - Michel Marcus, Mar 11 2022

A227217 Primes p such that p + (product of digits of p) is prime and p - (product of digits of p) is prime.

Original entry on oeis.org

23, 29, 83, 101, 103, 107, 109, 293, 307, 347, 349, 401, 409, 431, 439, 503, 509, 601, 607, 653, 659, 677, 701, 709, 743, 809, 907, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1123, 1201, 1297, 1301, 1303, 1307, 1409, 1423, 1489, 1523

Offset: 1

Derek Orr, Sep 19 2013

Intersection of A157677 and A225319.

Contains A056709. - Robert Israel, Apr 13 2015

			431 is prime, 431 + (4*3*1) = 443 is prime, and 431 - (4*3*1) = 419 is prime. So, 431 is a term in the sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007954, A056709, A157677, A225319.

filter:= proc(n) local m;
  if not isprime(n) then return false fi;
  m:= convert(convert(n,base,10),`*`);
  if m = 0 then return true fi;
  isprime(n+m) and isprime(n-m)
end proc:
select(filter, [seq(2*i+1,i=5..10000)]); # Robert Israel, Apr 13 2015

fQ[n_] := Block[{d = IntegerDigits@ n}, PrimeQ[n + Times @@ d] && PrimeQ[n - Times @@ d]]; Select[Prime@ Range@ 250, fQ] (* Michael De Vlieger, Apr 12 2015 *)

forprime(p=1,2000,d=digits(p);P=prod(i=1,#d,d[i]);if(isprime(p+P)&&isprime(p-P),print1(p,", "))) \\ Derek Orr, Apr 10 2015

from sympy import isprime, primerange
def DP(n):
    p = 1
    for i in str(n):
        p *= int(i)
    return p
for pn in primerange(1, 2000):
    dpn = DP(pn)
    if isprime(pn-dpn) and isprime(pn+dpn):
        print(pn, end=', ')
# Simplified by Derek Orr, Apr 10 2015

[p for p in primes_first_n(1000) if ((p-prod(Integer(p).digits(base=10))) in Primes() and (p+prod(Integer(p).digits(base=10))) in Primes())] # Tom Edgar, Sep 19 2013

More terms from Derek Orr, Apr 10 2015

A284290 Primes containing a digit 4.

Original entry on oeis.org

41, 43, 47, 149, 241, 347, 349, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 541, 547, 641, 643, 647, 743, 941, 947, 1049, 1249, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489

Offset: 1

Jaroslav Krizek, Mar 24 2017

Subsequence of A011534 and A062669.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. Primes containing a digit k for k = 0 - 9: A056709 (k = 0), A208270 (k = 1), A208272 (k = 2), A212525 (k = 3), A284290 (k = 4), A257667 (k = 5), A284291 (k = 6), A257668 (k = 7), A284292 (k = 8), A106093 (k = 9).

[p: p in PrimesUpTo(10000) | 4 in Intseq(p)]

Select[Range[1500], PrimeQ[#] && MemberQ[IntegerDigits[#], 4] &] (* Amiram Eldar, Nov 09 2019 *)

A284291 Primes containing a digit 6.

Original entry on oeis.org

61, 67, 163, 167, 263, 269, 367, 461, 463, 467, 563, 569, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 761, 769, 863, 967, 1061, 1063, 1069, 1163, 1361, 1367, 1567, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663

Offset: 1

Jaroslav Krizek, Mar 24 2017

Subsequence of A011536 and A062673.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Primes containing a digit k for k = 0 - 9: A056709 (k = 0), A208270 (k = 1), A208272 (k = 2), A212525 (k = 3), A284290 (k = 4), A257667 (k = 5), A284291 (k = 6), A257668 (k = 7), A284292 (k = 8), A106093 (k = 9).

[p: p in PrimesUpTo(10000) | 6 in Intseq(p)];

Select[Range[2000], PrimeQ[#] && MemberQ[IntegerDigits[#], 6] &] (* Amiram Eldar, Nov 09 2019 *)

A284292 Primes containing a digit 8.

Original entry on oeis.org

83, 89, 181, 281, 283, 383, 389, 487, 587, 683, 787, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 983, 1087, 1181, 1187, 1283, 1289, 1381, 1481, 1483, 1487, 1489, 1583, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867

Offset: 1

Jaroslav Krizek, Mar 25 2017

Subsequence of A011538 and A062677.

Differs from A062677 which contains also the composites 6889 = 83^2, 7387 = 83*89, 23489=83*283, 25187=89*283, 31789 = 83*383 etc. - R. J. Mathar, Mar 27 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. Primes containing a digit k for k = 0 - 9: A056709 (k = 0), A208270 (k = 1), A208272 (k = 2), A212525 (k = 3), A284290 (k = 4), A257667 (k = 5), A284291 (k = 6), A257668 (k = 7), this sequence (k = 8), A106093 (k = 9).

[p: p in PrimesUpTo(10000) | 8 in Intseq(p)];

isA284292 := proc(n)
    if isprime(n) then
        convert(convert(n,base,10),set) ;
        if 8 in % then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
for n from 1 to 2000 do
    if isA284292(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 27 2017

Select[Prime@ Range@ 500, MemberQ[ IntegerDigits@ #, 8] &] (* Giovanni Resta, Mar 25 2017 *)

from sympy import primerange
print([n for n in primerange(2, 2000) if '8' in str(n)]) # Indranil Ghosh, Mar 25 2017

cp's OEIS Frontend

A256186 Naught-y primes (A056709) that after removing all zeros become zeroless primes (A038618).

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A373294 a(n) is the number of n-digit primes that have at least one zero among their digits (A056709).

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A038618 Primes not containing the digit '0'.

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A134809 Cyclops primes.

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A157677 Primes p such that p + (product of digits of p) is also prime.

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Maple

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A101987 Product of nonzero digits of n-th prime.

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Maple

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A227217 Primes p such that p + (product of digits of p) is prime and p - (product of digits of p) is prime.

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