A045336 -id:A045336 - 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.

A083185 Palindromic primes using only nonprime digits (0,1,4,6,8,9).

Original entry on oeis.org

11, 101, 181, 191, 919, 10601, 11411, 16061, 16661, 18181, 18481, 19891, 19991, 91019, 94049, 94649, 94849, 94949, 96469, 98689, 1008001, 1114111, 1160611, 1180811, 1186811, 1190911, 1196911, 1409041, 1411141, 1444441, 1461641

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 26 2003

Harvey P. Dale, Table of n, a(n) for n = 1..100
P. De Geest, World!Of Palindromic Primes

Cf. A045336, A019546, A002385.

Palindromes in A034844.

Select[ Prime[ Range[111500]], IntegerDigits[ # ] == Reverse[ IntegerDigits[ # ]] && Union[ Join[ IntegerDigits[ # ], {0, 1, 4, 6, 8, 9}]] == {0, 1, 4, 6, 8, 9} & ]
Select[Prime[Range[120000]],PalindromeQ[#]&&NoneTrue[IntegerDigits[#], PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 08 2018 *)

Edited and extended by Patrick De Geest, Jun 11 2003

A083183 Smallest palindromic prime using only prime digits (2,3,5,7) and having a sum of digits = prime(n), or 0 if no such number exists.

Original entry on oeis.org

2, 3, 5, 7, 353, 373, 33533, 757, 35753, 75557, 77377, 3773773, 325777523, 327757723, 357575753, 32577577523, 32777777723, 35775757753, 72777777727, 3557577757553, 3777727277773, 3775777775773, 7777573757777

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 26 2003

Conjecture: no entry is zero.

Chai Wah Wu, Table of n, a(n) for n = 1..39

Cf. A083184, A045336.

Corrected and extended by Patrick De Geest, Jun 12 2003

A343714 Palindromic primes of the form p//q//reverse(p), where p is a prime (not necessarily palindromic) and q, of course, is a palindromic prime.

Original entry on oeis.org

353, 373, 727, 757, 11311, 13331, 19391, 31013, 31513, 33533, 37273, 37573, 39293, 71317, 71917, 73237, 77977, 79397, 97379, 97579, 1035301, 1092901, 1093901, 1175711, 1178711, 1273721, 1317131, 1335331, 1338331, 1513151, 1572751, 1633361, 1737371, 1793971

Offset: 1

Jon E. Schoenfield, May 08 2021

Note that reverse(p) need not be a prime; e.g., a(7)=19391 is the concatenation of 19, 3, and 91=7*13. If a requirement were added that reverse(p) also be a prime, the result would be sequence A343715.

			353 is a term because it is a palindromic prime (A002385) and is the concatenation of 3 (a prime), 5 (a palindromic prime), and 3 (the reverse of 3).
31513 is a term in two ways: as the concatenation 3//151//3 and as the concatenation 31//5//13.
7392937 is a term in three ways: 7//39293//7, 73//929//37, and 739//2//937.

Cf. A002385, A045336, A177678, A343715.

A343715 Palindromic primes of the form p//q//reverse(p), where p, q, and reverse(p) are primes.

Original entry on oeis.org

353, 373, 727, 757, 11311, 13331, 31013, 31513, 33533, 37273, 37573, 39293, 71317, 71917, 73237, 77977, 79397, 97379, 97579, 1175711, 1178711, 1317131, 1335331, 1338331, 1513151, 1572751, 1737371, 1793971, 1917191, 1993991, 1995991, 3103013, 3106013, 3127213

Offset: 1

Jon E. Schoenfield, May 08 2021

If reverse(p) were allowed to be nonprime, the result would be sequence A343714, which includes such terms as 19391.

			353 is a term because it is a palindromic prime (A002385) and is the concatenation of 3 (a prime), 5 (a palindromic prime), and 3 (the reverse of 3, and also a prime).
31513 is a term in two ways: as the concatenation 3//151//3 and as the concatenation 31//5//13.
7392937 is a term in three ways: 7//39293//7, 73//929//37, and 739//2//937.

Cf. A002385, A045336, A177678, A343714.

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

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A083185 Palindromic primes using only nonprime digits (0,1,4,6,8,9).

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A083183 Smallest palindromic prime using only prime digits (2,3,5,7) and having a sum of digits = prime(n), or 0 if no such number exists.

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A343714 Palindromic primes of the form p//q//reverse(p), where p is a prime (not necessarily palindromic) and q, of course, is a palindromic prime.

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A343715 Palindromic primes of the form p//q//reverse(p), where p, q, and reverse(p) are primes.

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