A019546 Primes whose digits are primes.

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

1,1

Comments

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

References

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

Links

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

Mathematica

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 *)

Prog

(PARI) is_A019546(n)=isprime(n) & !setminus(Set(Vec(Str(n))), Vec("2357")) \\ M. F. Hasler, Jan 13 2012
(PARI) 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
(Haskell)
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
(Magma) [p: p in PrimesUpTo(5600) | Set(Intseq(p)) subset [2, 3, 5, 7]]; // Bruno Berselli, Jan 13 2012
(Python)
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

Crossrefs

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

Sequence in context: A074491 A154385 A125525 * A104179 A096148 A211681

Adjacent sequences: A019543 A019544 A019545 * A019547 A019548 A019549

Keyword

nonn,base

Author

R. Muller

Extensions

More terms from Cino Hilliard, Aug 06 2006

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

Status

approved