A020463 -id:A020463 - 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.

A020458 Primes that contain digits 2 and 3 only.

Original entry on oeis.org

2, 3, 23, 223, 233, 2333, 3323, 23333, 32233, 32323, 33223, 222323, 232333, 233323, 323233, 323333, 333233, 333323, 2222333, 2223233, 2232323, 2233223, 2332333, 2333323, 3222223, 3223223, 3223333, 3233323, 3233333, 3332233, 3333233, 22222223, 22223323, 22232233

Offset: 1

David W. Wilson

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)

Cf. A139067, A139069, A032810, A020463.

Flatten[Table[Select[FromDigits/@Tuples[{2,3},n],PrimeQ],{n,7}]] (* Harvey P. Dale, Jul 13 2012 *)

go(n)=my(v=List([2]),x,t); for(d=1,n, x=10^d\9*2; forstep(i=1,2^d-1,2, if(ispseudoprime(t=x+fromdigits(binary(i))), listput(v,t)))); Vec(v) \\ Charles R Greathouse IV, Sep 14 2015

from sympy import isprime
from itertools import count, islice, product
def agen(): # generator of terms
    yield from [2, 3]
    for d in count(2):
        for first in product("23", repeat=d-1):
            t = int("".join(first) + "3")
            if isprime(t): yield t
print(list(islice(agen(), 34))) # Michael S. Branicky, Jun 08 2022

Edited by N. J. A. Sloane, Jul 27 2008 at the suggestion of Dmitry Kamenetsky.

Edited by Charles R Greathouse IV, Mar 17 2010

A260378 Primes having only {0, 3, 7} as digits.

Original entry on oeis.org

3, 7, 37, 73, 307, 337, 373, 733, 773, 3037, 3307, 3373, 3733, 7307, 7333, 7703, 30307, 30703, 30707, 30773, 33037, 33073, 33377, 33703, 33773, 37003, 37307, 37337, 70003, 70373, 73037, 73303, 77003, 77377, 77773, 300007, 300073, 300733, 303007, 303073, 303307

Offset: 1

Vincenzo Librandi, Jul 24 2015

A020463 is a subsequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index to entries for primes with digits in a given set

Cf. Primes that contain only the digits (k,3,7): this sequence (k=0), A260379 (k=1), A214704 (k=2), A199347 (k=4), A087363 (k=5), A260380 (k=6), A260381 (k=8), A260382 (k=9).
Subsequence of A155055.
Cf. A000040, A020463.

[p: p in PrimesUpTo(5*10^5) | Set(Intseq(p)) subset [0, 3, 7]];

Select[Prime[Range[4 10^4]], Complement[IntegerDigits[#],{0, 3, 7}]=={} &]

A260379 Primes having only {1, 3, 7} as digits.

Original entry on oeis.org

3, 7, 11, 13, 17, 31, 37, 71, 73, 113, 131, 137, 173, 311, 313, 317, 331, 337, 373, 733, 773, 1117, 1171, 1373, 1733, 1777, 3137, 3313, 3331, 3371, 3373, 3733, 7177, 7331, 7333, 7717, 11113, 11117, 11131, 11171, 11173, 11177, 11311, 11317, 11717, 11731, 11777

Offset: 1

Vincenzo Librandi, Jul 24 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index to entries for primes with digits in a given set

Subsequence of A030096 and A155055. A020451, A020455, and A020463 are subsequences.
Cf. similar sequences listed in A260378.
Cf. A000040, A020451, A020455, A020463.

[p: p in PrimesUpTo(2*10^4) | Set(Intseq(p)) subset [1, 3, 7]];

Select[Prime[Range[2 10^3]], Complement[IntegerDigits[#], {1, 3, 7}]=={} &]

A260380 Primes having only {3, 6, 7} as digits.

Original entry on oeis.org

3, 7, 37, 67, 73, 337, 367, 373, 673, 677, 733, 773, 3373, 3637, 3673, 3677, 3733, 3767, 6337, 6367, 6373, 6637, 6673, 6733, 6737, 6763, 7333, 7673, 33377, 33637, 33767, 33773, 36373, 36637, 36677, 36767, 37337, 37363, 37633, 37663, 63337, 63367, 63377, 63667

Offset: 1

Vincenzo Librandi, Aug 01 2015

A020463 and A020469 are subsequences.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index to entries for primes with digits in a given set

Cf. similar sequences listed in A260378.

Cf. A020463, A020469.

[p: p in PrimesUpTo(2*10^5) | Set(Intseq(p)) subset [3, 6, 7]];

Select[Prime[Range[2 10^4]], Complement[IntegerDigits[#], {3, 6, 7}]=={} &]
Select[Flatten[Table[FromDigits/@Tuples[{3,6,7},n],{n,5}]],PrimeQ] (* Harvey P. Dale, Apr 13 2022 *)

A260381 Primes having only {3, 7, 8} as digits.

Original entry on oeis.org

3, 7, 37, 73, 83, 337, 373, 383, 733, 773, 787, 877, 883, 887, 3373, 3733, 3833, 3877, 7333, 7873, 7877, 7883, 8377, 8387, 8737, 8783, 8837, 8887, 33377, 33773, 37337, 37783, 38333, 38377, 38737, 38783, 38833, 38873, 73387, 73783, 73877, 73883, 77377, 77383

Offset: 1

Vincenzo Librandi, Aug 01 2015

A020463, A020464 and A020470 are subsequences.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index to entries for primes with digits in a given set

Cf. similar sequences listed in A260378.

Cf. A000040, A020463, A020464, A020470.

[p: p in PrimesUpTo(2*10^5) | Set(Intseq(p)) subset [3, 7, 8]];

Select[Prime[Range[2 10^4]], Complement[IntegerDigits[#], {3, 7, 8}]=={} &]

A260382 Primes having only {3, 7, 9} as digits.

Original entry on oeis.org

3, 7, 37, 73, 79, 97, 337, 373, 379, 397, 733, 739, 773, 797, 937, 977, 997, 3373, 3733, 3739, 3779, 3793, 3797, 7333, 7393, 7793, 7933, 7937, 7993, 9337, 9377, 9397, 9733, 9739, 9973, 33377, 33739, 33773, 33797, 33937, 33997, 37337, 37339, 37379, 37397

Offset: 1

Vincenzo Librandi, Aug 01 2015

A020463 and A020471 are subsequences.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)
Index to entries for primes with digits in a given set

Cf. similar sequences listed in A260378.

Cf. A000040, A020463, A020471.

[p: p in PrimesUpTo(2*10^5) | Set(Intseq(p)) subset [3,7,9]];

Select[Prime[Range[5*10^3]], Complement[IntegerDigits[#], {3, 7, 9}]=={} &]

A143967 Numbers containing only digits 3 or 7 in decimal representation.

Original entry on oeis.org

3, 7, 33, 37, 73, 77, 333, 337, 373, 377, 733, 737, 773, 777, 3333, 3337, 3373, 3377, 3733, 3737, 3773, 3777, 7333, 7337, 7373, 7377, 7733, 7737, 7773, 7777, 33333, 33337, 33373, 33377, 33733, 33737, 33773, 33777, 37333, 37337, 37373, 37377

Offset: 1

Reinhard Zumkeller, Sep 06 2008

See A020463 for primes.

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

Cf. A004086, A020463, A032810.

a143967 = f 0 . (+ 1) where
   f y 1 = a004086 y
   f y x = f (10 * y + 3 + 4 * r) x' where (x', r) = divMod x 2
-- Reinhard Zumkeller, Mar 18 2015

Table[FromDigits/@Tuples[{3,7},n],{n,5}]//Flatten (* Harvey P. Dale, Aug 28 2017 *)

a(n) = f(n+1, 0) with f(n, x) = if n=1 then A004086(x) else f(floor(n/2), 10*x + 3 + 4*(n mod 2)).

A036316 Composite numbers whose prime factors contain no digits other than 3 and 7.

Original entry on oeis.org

9, 21, 27, 49, 63, 81, 111, 147, 189, 219, 243, 259, 333, 343, 441, 511, 567, 657, 729, 777, 999, 1011, 1029, 1119, 1323, 1369, 1533, 1701, 1813, 1971, 2187, 2199, 2319, 2331, 2359, 2401, 2611, 2701, 2997, 3033, 3087, 3357, 3577, 3969, 4107, 4599, 5103

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020463. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 800 terms from Vincenzo Librandi)
Index entries for sequences related to prime factors.

Cf. A003594, A020463, A036302-A036325.

[n: n in [9..6000] | not IsPrime(n) and forall{f: f in PrimeDivisors(n) | Intseq(f) subset [3,7]}]; // Bruno Berselli, Aug 26 2013

dpfQ[n_]:=Module[{d=Union[Flatten[IntegerDigits/@Transpose[FactorInteger[n]][[1]]]]}, !PrimeQ[n]&&(d == {3}||d == {7}||d == {3, 7})]; Select[Range[6000], dpfQ] (* Vincenzo Librandi, Aug 25 2013 *)

Sum_{n>=1} 1/a(n) = Product_{p in A020463} (p/(p - 1)) - Sum_{p in A020463} 1/p - 1 = 0.3143000293... . - Amiram Eldar, May 22 2022

A036942 Smallest n-digit prime containing only digits 3 and 7, or 0 if no such prime exists.

Original entry on oeis.org

3, 37, 337, 3373, 33377, 333337, 3333373, 33333373, 333337777, 3333733373, 33333333377, 333333733333, 3333333377377, 33333333373777, 333333333337337, 3333333333373777, 33333333333337337, 333333333333337333

Offset: 1

Patrick De Geest, Jan 04 1999

Jinyuan Wang, Table of n, a(n) for n = 1..500

Cf. A020463, A036229, A036316.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

Python

Extensions

A020458 Primes that contain digits 2 and 3 only.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A260378 Primes having only {0, 3, 7} as digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A260379 Primes having only {1, 3, 7} as digits.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A260380 Primes having only {3, 6, 7} as digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A260381 Primes having only {3, 7, 8} as digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A260382 Primes having only {3, 7, 9} as digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A143967 Numbers containing only digits 3 or 7 in decimal representation.

Views

Author