A179336 -id:A179336 - cp's OEIS frontend

A019546 Primes whose digits are primes; primes having only {2, 3, 5, 7} as digits.

2, 3, 5, 7, 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 2237, 2273, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3373, 3527, 3533, 3557, 3727, 3733, 5227, 5233, 5237, 5273, 5323, 5333, 5527, 5557

Offset: 1

R. Muller

Intersection of A046034 and A000040; A055642(a(n)) = A193238(a(n)). - Reinhard Zumkeller, Jul 19 2011
Ribenboim mentioned in 2000 the following number as largest known term: 72323252323272325252 * (10^3120 - 1) / (10^20 - 1) + 1. It has 3120 digits, and was discovered by Harvey Dubner in 1992. Larger terms are 22557252272*R(15600)/R(10) and 2255737522*R(15600) where R(n) denotes the n-th repunit (see A002275): Both have 15600 digits and were found in 2002, also by Dubner (see Weisstein link). David Broadhurst reports in 2003 an even longer number with 82000 digits: (10^40950+1) * (10^20055+1) * (10^10374 + 1) * (10^4955 + 1) * (10^2507 + 1) * (10^1261 + 1) * (3*R(1898) + 555531001*10^940 - R(958)) + 1, see link. - Reinhard Zumkeller, Jan 13 2012
The smallest and largest primes that use exactly once the four prime decimal digits are respectively a(27)= 2357 and a(54) = 7523. - Bernard Schott, Apr 27 2023

Paulo Ribenboim, Prime Number Records (Chap 3), in 'My Numbers, My Friends', Springer-Verlag 2000 NY, page 76.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
József Bölcsföldi and György Birkás, Golden ratio prime numbers, International Journal of Engineering Science Invention (2018) Vol. 6 Issue 12, 82-85.
David Broadhurst: primeform, 82000-digit prime with all digits prime [Broken link]
David Broadhurst, 82000-digit prime with all digits prime, digest of 2 messages in primeform Yahoo group, Oct 20 - Oct 25, 2003.
Chris K. Caldwell, The Prime Glossary: Prime-digit prime
Chris K. Caldwell and G. L. Honaker, Jr., 2357, Prime Curios!
Chris K. Caldwell and G. L. Honaker, Jr., 7523, Prime Curios!
H. Ibstedt, A Few Smarandache Integer Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, pp. 171-183.
Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.
Eric Weisstein's MathWorld Headline News, Two Gigantic Primes with Prime Digits Found
Eric Weisstein's World of Mathematics, Smarandache Sequences
Index to entries for primes with digits in a given set

Cf. A045336, A003631, A034844, A179336, A109066, A215927.
Cf. A020463 (subsequence).
A093162, A093164, A093165, A093168, A093169, A093672, A093674, A093675, A093938 and A093941 are subsequences. - XU Pingya, Apr 20 2017

a019546 n = a019546_list !! (n-1)
a019546_list = filter (all (`elem` "2357") . show )
                      ([2,3,5] ++ (drop 2 a003631_list))
-- Or, much more efficient:
a019546_list = filter ((== 1) . a010051) $
                      [2,3,5,7] ++ h ["3","7"] where
   h xs = (map read xs') ++ h xs' where
     xs' = concat $ map (f xs) "2357"
     f xs d = map (d :) xs
-- Reinhard Zumkeller, Jul 19 2011

[p: p in PrimesUpTo(5600) | Set(Intseq(p)) subset [2,3,5,7]]; // Bruno Berselli, Jan 13 2012

Select[Prime[Range[700]], Complement[IntegerDigits[#], {2, 3, 5, 7}] == {} &] (* Alonso del Arte, Aug 27 2012 *)
Select[Prime[Range[700]], AllTrue[IntegerDigits[#], PrimeQ] &] (* Ivan N. Ianakiev, Jun 23 2018 *)
Select[Flatten[Table[FromDigits/@Tuples[{2,3,5,7},n],{n,4}]],PrimeQ] (* Harvey P. Dale, Apr 05 2025 *)

is_A019546(n)=isprime(n) & !setminus(Set(Vec(Str(n))),Vec("2357")) \\ M. F. Hasler, Jan 13 2012

print1(2); for(d=1,4, forstep(i=1,4^d-1,[1,1,2], p=sum(j=0,d-1,10^j*[2,3,5,7][(i>>(2*j))%4+1]); if(isprime(p), print1(", "p)))) \\ Charles R Greathouse IV, Apr 29 2015

from itertools import product
from sympy import isprime
A019546_list = [2,3,5,7]+[p for p in (int(''.join(d)+e) for l in range(1,5) for d in product('2357',repeat=l) for e in '37') if isprime(p)] # Chai Wah Wu, Jun 04 2021

More terms from Cino Hilliard, Aug 06 2006

Thanks to Charles R Greathouse IV and T. D. Noe for massive editing support.

A080608 Deletable primes: primes such that removing some digit leaves either the empty string or another deletable prime (possibly preceded by some zeros).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 103, 107, 113, 127, 131, 137, 139, 157, 163, 167, 173, 179, 193, 197, 223, 229, 233, 239, 263, 269, 271, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 397, 431, 433, 439

Offset: 1

David W. Wilson, Feb 25 2003

Subsequence of A179336. - Reinhard Zumkeller, Jul 11 2010
Leading zeros are allowed in the number that appears after the digit is deleted. For example the prime 100003 is deletable because of the sequence 00003, 0003, 003, 03, 3 consists of primes. Because of this, it appears that deletable primes are relatively common in the region just above a power of ten. For example 10^1000 + 2713 is a deletable prime. - Jeppe Stig Nielsen, Aug 01 2018
For a version that does not allow leading zeros, see A305352. - Jeppe Stig Nielsen, Aug 01 2018

			410256793 is a deletable prime since each member of the sequence 410256793, 41256793, 4125673, 415673, 45673, 4567, 467, 67, 7 is prime (Weisstein, Caldwell).

David W. Wilson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Deletable Prime

Cf. A080603, A096235-A096246, A305352.

read("transforms"):
isA080608 := proc(n)
    option remember;
    local dgs,i ;
    if isprime(n) then
        if n < 10 then
            true;
        else
            dgs := convert(n,base,10) ;
            for i from 1 to nops(dgs) do
                subsop(i=NULL,dgs) ;
                digcatL(ListTools[Reverse](%)) ;
                if procname(%)  then
                    return true;
                end if;
            end do:
            false ;
        end if;
    else
        false;
    end if;
end proc:
n := 1;
for i from 1 to 500 do
    p := ithprime(i) ;
    if isA080608(p) then
        printf("%d %d\n",n,p) ;
        n := n+1 ;
    fi ;
end do: # R. J. Mathar, Oct 11 2014

Rest@ Union@ Nest[Function[{a, p}, Append[a, With[{w = IntegerDigits[p]}, If[# == True, p, 0] &@ AnyTrue[Array[FromDigits@ Delete[w, #] &, Length@ w], ! FreeQ[a, #] &]]]] @@ {#, Prime[Length@ # + 1]} &, Prime@ Range@ PrimePi@ 10, 81] (* Michael De Vlieger, Aug 02 2018 *)

is(n) = !ispseudoprime(n)&&return(0);my(d=digits(n));#d==1&&return(1);for(i=1,#d,is(fromdigits(vecextract(d,Str("^"i))))&&return(1));0 \\ Jeppe Stig Nielsen, Aug 01 2018

use ntheory ":all"; sub is { my $n=shift; return 0 unless is_prime($n); my @d=todigits($n); return 1 if @d==1; is(fromdigits([vecextract(\@d,~(1<<$))])) && return 1 for 0..$#d; 0; } # _Dana Jacobsen, Nov 16 2018

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    if n < 10: return True
    s = str(n)
    si = (s[:i]+s[i+1:] for i in range(len(s)))
    return any(ok(int(t)) for t in si)
print([k for k in range(440) if ok(k)]) # Michael S. Branicky, Jan 28 2023

A034844 Primes with only nonprime decimal digits.

Original entry on oeis.org

11, 19, 41, 61, 89, 101, 109, 149, 181, 191, 199, 401, 409, 419, 449, 461, 491, 499, 601, 619, 641, 661, 691, 809, 811, 881, 911, 919, 941, 991, 1009, 1019, 1049, 1061, 1069, 1091, 1109, 1181, 1409, 1481, 1489, 1499, 1601, 1609, 1619, 1669, 1699, 1801, 1811

Offset: 1

Erich Friedman

A109066(n) = 0 iff prime(n) is in this sequence. [Reinhard Zumkeller, Jul 11 2010, corrected by M. F. Hasler, Aug 27 2012]
Or, primes p such that A193238(p) = 0. - M. F. Hasler, Aug 27 2012
Intersection of A084984 and A000040; complement of A179336 (within the primes A000040). [Reinhard Zumkeller, Jul 19 2011, edited by M. F. Hasler, Aug 27 2012]
The smallest prime that contains all the six nonprime decimal digits is a(694) = 104869 (see Prime Curios! link). - Bernard Schott, Mar 21 2023

			E.g. 149 is a prime made of nonprime digits(1,4,9).
991 is a prime without any prime digits.

T. D. Noe, Table of n, a(n) for n = 1..1000
Chris Caldwell, The First 1,000 Primes
Chris K. Caldwell and G. L. Honaker, Jr., 104869, Prime Curios!

Cf. A000040, A084984.

Cf. A033274, A109066, A152313, A141409.

a034844 n = a034844_list !! (n-1)
a034844_list = filter (not . any  (`elem` "2357") . show ) a000040_list
-- Reinhard Zumkeller, Jul 19 2011

[p: p in PrimesUpTo(2000) | forall{d: d in [2,3,5,7] | d notin Set(Intseq(p))}];  // Bruno Berselli, Jul 27 2011

Select[Prime[Range[279]], Intersection[IntegerDigits[#], {2, 3, 5, 7}] == {} &] (* Jayanta Basu, Apr 18 2013 *)
Union[Select[Flatten[Table[FromDigits/@Tuples[{1,4,6,8,9,0},n],{n,2,4}]],PrimeQ]] (* Harvey P. Dale, Dec 08 2014 *)

is_A034844(n)=isprime(n)&!apply(x->isprime(x),eval(Vec(Str(n)))) \\ M. F. Hasler, Aug 27 2012

is_A034844(n)=isprime(n)&!setintersect(Set(Vec(Str(n))),Vec("2357")) \\ M. F. Hasler, Aug 27 2012

from sympy import isprime
from itertools import product
def auptod(maxdigits):
    alst = []
    for d in range(1, maxdigits+1):
        for p in product("014689", repeat=d-1):
            if d > 1 and p[0] == "0": continue
            for end in "19":
                s = "".join(p) + end
                t = int(s)
                if isprime(t): alst.append(t)
    return alst
print(auptod(4)) # Michael S. Branicky, Nov 19 2021

a(n) >> n^1.285. [Charles R Greathouse IV, Feb 20 2012]

Edited by N. J. A. Sloane, Feb 22 2009 at the suggestion of R. J. Mathar

A109066 Number of prime digits in n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 2, 1, 0, 1, 2, 1, 1, 2, 1, 0, 0, 1, 1, 0, 1, 3, 3, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 2, 2, 3, 2, 2, 1, 2, 0, 0, 0, 1, 1, 2, 1, 1, 0, 2, 0, 1, 1, 1, 1, 0, 0, 2, 1, 2, 3, 1, 2, 3, 2, 1, 2

Offset: 1

Zak Seidov, Jun 17 2005

The prime A000040(n) is in A034844 iff a(n) = 0; it is in A179336 iff a(n) > 0. [Reinhard Zumkeller, Jul 11 2010, corrected by M. F. Hasler, Aug 27 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A019546 (primes whose digits are primes), A092629 (number of prime digits is nonprime), A104250 (sum of prime digits of n-th prime).

Cf. A000040, A034844, A179336.

a[n_]:=Count[PrimeQ/@IntegerDigits[Prime[n]], True]

a(n) = vecsum(apply(x->isprime(x), digits(prime(n)))); \\ Michel Marcus, Mar 15 2019

a(n) = A193238(A000040(n)). [Reinhard Zumkeller, Jul 19 2011]

A118950 Numbers containing at least one prime digit.

Original entry on oeis.org

2, 3, 5, 7, 12, 13, 15, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 45, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 62, 63, 65, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 112

Offset: 1

Rick L. Shepherd, May 06 2006

A193238(a(n)) > 0; complement of A084984; A092620, A092624 and A092625 are subsequences. - Reinhard Zumkeller, Jul 19 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A118951, A092620, A084984, A011540, A011531, A046034, A179336 (primes).

a118950 n = a118950_list !! (n-1)
a118950_list = filter (any (`elem` "2357") . show ) [0..]
-- Reinhard Zumkeller, Jul 19 2011

Select[Range[150],AnyTrue[IntegerDigits[#],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 19 2018 *)

is(n)=!!#select(isprime, digits(n)) \\ Charles R Greathouse IV, Sep 15 2015

a(n) = n + O(n^k) with k = log 6/log 10 = 0.77815.... - Charles R Greathouse IV, Sep 15 2015

A179335 a(n) is the smallest prime which appears as a substring of the decimal representation of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 3, 7, 19, 2, 2, 3, 3, 41, 3, 7, 3, 5, 61, 7, 7, 3, 7, 3, 89, 7, 101, 3, 7, 109, 3, 2, 3, 3, 3, 149, 5, 5, 3, 7, 3, 7, 181, 19, 3, 7, 19, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 401, 409, 19, 2, 3, 3, 3, 3, 449, 5, 61, 3, 7, 7, 7

Offset: 1

Reinhard Zumkeller, Jul 11 2010

a(n) < 10 iff prime(n) is in A179336;

a(n) = prime(n) iff prime(n) is in A033274. [Corrected by M. F. Hasler, Aug 27 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A070024, A019546, A092629, A104250, A019546, A034844, A179336, A193238.

A179335(n)={my(p=prime(n),m=0,M); for(d=1,n, M=10^d; n=p; until(n<=M || !n\=10, isprime(n%M) & (!m || m>n%M) & m=n%M); m & return(m))} \\ M. F. Hasler, Aug 27 2012

from sympy import isprime, prime
def a(n):
    s = str(prime(n))
    ss = set(int(s[i:i+1+l]) for i in range(len(s)) for l in range(len(s)))
    return min(t for t in ss if isprime(t))
print([a(n) for n in range(1, 94)]) # Michael S. Branicky, Jun 29 2022

A108614 Semiprimes with non-semiprimes digits (no digits 4,6,9 in semiprimes).

Original entry on oeis.org

10, 15, 21, 22, 25, 33, 35, 38, 51, 55, 57, 58, 77, 82, 85, 87, 111, 115, 118, 121, 122, 123, 133, 155, 158, 177, 178, 183, 185, 187, 201, 202, 203, 205, 213, 215, 217, 218, 221, 235, 237, 253, 278, 287, 301, 302, 303, 305, 321, 323, 327, 335, 355, 358, 371

Offset: 1

Zak Seidov, Jun 13 2005

Complement of A107342 in the class of semiprimes.

This is to semiprimes A001358 as A034844 (Primes with nonprime digits) is to primes A000040. [Jonathan Vos Post, Jul 15 2010]

Cf. A107342, A137167.

Cf. A107665, A107666, A111494, A111496, A111697, A179336. [Jonathan Vos Post, Jul 15 2010]

cnd[n_]:=Plus@@Last/@FactorInteger[n]==2&&Union[FreeQ[IntegerDigits[n], # ]&/@{4, 6, 9}]=={True};Select[Range[600], cnd[ # ]&]

{j in A001358 and j not in A179463}. [Jonathan Vos Post, Jul 15 2010]

A179463 Semiprimes A001358 containing at least one semiprime digit in base 10.

Original entry on oeis.org

4, 6, 9, 14, 26, 34, 39, 46, 49, 62, 65, 69, 74, 86, 91, 93, 94, 95, 106, 119, 129, 134, 141, 142, 143, 145, 146, 159, 161, 166, 169, 194, 206, 209, 214, 219, 226, 247, 249, 254, 259, 262, 265, 267, 274, 289, 291, 295, 298, 299, 309, 314, 319, 326, 329, 334, 339, 341

Offset: 1

Jonathan Vos Post, Jul 15 2010

Semiprimes containing at least one 4, 6, or 9 digit base 10.
This is to semiprimes A001358 as A179336 is to primes A000040.
This properly includes the subset A107342 Semiprimes with semiprime digits.

Cf. A107342 Semiprimes with semiprime digits (digits 4, 6, 9 only), A107665 Numbers with semiprime digits (digits 4, 6, 9 only), A107666 Primes with semiprime digits (digits 4, 6, 9 only), A111494, A111496, A111697, A108614 Semiprimes with non-semiprimes digits (no digits 4, 6, 9 in semiprimes), A179336.

spdQ[n_]:=Module[{idn=IntegerDigits[n]},MemberQ[idn,4] || MemberQ[ idn,6] || MemberQ[ idn,9]]; Select[Select[Range[350],PrimeOmega[#]==2&],spdQ] (* Harvey P. Dale, Jun 24 2013 *)

Corrected (a(37) added) by Harvey P. Dale, Jun 24 2013

cp's OEIS Frontend

A019546 Primes whose digits are primes; primes having only {2, 3, 5, 7} as digits.

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A080608 Deletable primes: primes such that removing some digit leaves either the empty string or another deletable prime (possibly preceded by some zeros).

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A034844 Primes with only nonprime decimal digits.

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A109066 Number of prime digits in n-th prime.

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A118950 Numbers containing at least one prime digit.

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A179335 a(n) is the smallest prime which appears as a substring of the decimal representation of prime(n).

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