A109066 -id:A109066 - 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.

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

A033274 Primes that do not contain any other prime as a proper substring.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 41, 61, 89, 101, 109, 149, 181, 401, 409, 449, 491, 499, 601, 691, 809, 881, 991, 1009, 1049, 1069, 1481, 1609, 1669, 1699, 1801, 4001, 4049, 4481, 4649, 4801, 4909, 4969, 6091, 6469, 6481, 6869, 6949, 8009, 8069, 8081, 8609, 8669, 8681

Offset: 1

Michael Kleber

If there is more than one digit, all digits must be nonprime numbers.
A179335(n) = prime(n) iff prime(n) is in this sequence. For n > 4, prime(n) is in this sequence iff A109066(n) = 0. - Reinhard Zumkeller, Jul 11 2010, corrected by M. F. Hasler, Aug 27 2012
A079066(n) = 0 iff prime(n) is in this sequence. [Corrected by M. F. Hasler, Aug 27 2012]
What are the asymptotics of this sequence? - Charles R Greathouse IV, Aug 27 2012

			149 is a term as 1, 4, 9, 14, 49 are all nonprimes.
199 is not a term as 19 is a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)

Cf. A089768, A089770, A039996, A079397, A033274, A034844, A179909-A179919.

import Data.List (elemIndices)
a033274 n = a033274_list !! (n-1)
a033274_list = map (a000040 . (+ 1)) $ elemIndices 0 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 1100, f@# == 1 &] (* Robert G. Wilson v, Aug 01 2010 *)
Select[Prime[Range[1100]],NoneTrue[Flatten[Table[FromDigits/@Partition[IntegerDigits[#],d,1],{d,IntegerLength[#]-1}]],PrimeQ]&] (* Harvey P. Dale, Apr 19 2025 *)

from sympy import isprime
def ok(n):
    if n in {2, 3, 5, 7}: return True
    s = str(n)
    if set(s) & {"2", "3", "5", "7"} or not isprime(n): return False
    ss2 = set(s[i:i+l] for i in range(len(s)-1) for l in range(2, len(s)))
    return not any(isprime(int(ss)) for ss in ss2)
print([k for k in range(9000) if ok(k)]) # Michael S. Branicky, Jun 29 2022

Edited by N. J. A. Sloane at the suggestion of Luca Colucci, Apr 03 2008

A193238 Number of prime digits in decimal representation of n.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Jul 19 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A010051.

Cf. A211681, A046034, A052382, A055640, A055641, A055642, A102669 - A102685, A122640, A117804, A196563, A196564.

a193238 n = length $ filter (`elem` "2357") $ show n

Count[IntegerDigits[#],?PrimeQ]&/@Range[0,100] (* _Harvey P. Dale, Nov 13 2022 *)

a(n)=n=eval(Vec(Str(n)));sum(i=1,#n,isprime(n[i])) \\ Charles R Greathouse IV, Jul 29 2011

a(A084984(n))=0; a(A118950(n))>0; a(A092620(n))=1; a(A092624(n))=2; a(A092625(n))=3; a(A046034(n))=A055642(A046034(n));
a(A000040(n)) = A109066(n).
From Hieronymus Fischer, May 30 2012: (Start)
a(n) = sum_{j=1..m+1} (floor(n/10^j+0.3) + floor(n/10^j+0.5) + floor(n/10^j+0.8) - floor(n/10^j+0.2) - floor(n/10^j+0.4) - floor(n/10^j+0.6)), where m=floor(log_10(n)), n>0.
a(10n+k) = a(n) + a(k), 0<=k<10, n>=0.
a(n) = a(floor(n/10)) + a(n mod 10), n>=0.
a(n) = sum_{j=0..m} a(floor(n/10^j) mod 10), n>=0.
a(A046034(n)) = floor(log_4(3n+1)), n>0.
a(A211681(n)) = 1 + floor((n-1)/4), n>0.
G.f.: g(x) = (1/(1-x))*sum_{j>=0} (x^(2*10^j) + x^(3*10^j)+ x^(5*10^j) + x^(7*10^j))*(1-x^10^j)/(1-x^10^(j+1)).
Also: g(x) = (1/(1-x))*sum_{j>=0} (x^(2*10^j)- x^(4*10^j)+ x^(5*10^j)- x^(6*10^j)+ x^(7*10^j)- x^(8*10^j))/(1-x^10^(j+1)). (End)

A179336 Primes containing at least one prime digit in base 10.

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, 151, 157, 163, 167, 173, 179, 193, 197, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Reinhard Zumkeller, Jul 11 2010

a(n) = A080608(n) for n<28; A080608 is a subsequence;
A179335(n) < 10 iff prime(n) is in this sequence;
A109066(n) > 0 iff prime(n) is in this sequence. [Corrected by M. F. Hasler, Aug 27 2012]

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

Intersection of A118950 and A000040; relative complement A000040 \ A034844.

Cf. A019546, A108419.

a179336 n = a179336_list !! (n-1)
a179336_list = filter (any (`elem` "2357") . show ) a000040_list
-- Reinhard Zumkeller, Jul 19 2011

a(n) ~ n log n. - Charles R Greathouse IV, Nov 01 2022

A155548 Primes p such that p and the p-th prime have the same number of prime digits.

Original entry on oeis.org

2, 3, 7, 17, 37, 47, 73, 83, 89, 97, 113, 163, 179, 193, 197, 251, 347, 359, 383, 397, 421, 431, 443, 487, 541, 547, 571, 593, 607, 617, 631, 653, 673, 677, 719, 727, 743, 751, 761, 787, 821, 829, 857, 877, 881, 883, 947, 971, 1009, 1013, 1019, 1021, 1051, 1087

Offset: 1

Juri-Stepan Gerasimov, Jan 24 2009

Prime digit = 2, 3, 5 or 7.

			2 is a term because 2 is prime, prime(2)=3, and 2 and 3 each have 1 prime digit.
3 is a term because 3 is prime, prime(3)=5, and 3 and 5 each have 1 prime digit.
4 is not a term because 4 is not prime.
5 is prime, but prime(5)=11, and 5 has 1 prime digit while 11 has 0 prime digits, so 5 is not a term.

Cf. A000040, A109066.

mm := proc (m) options operator, arrow: convert(m, base, 10) end proc: a := proc (n) local t, s, j: t := 0: s := 0: for j to nops(mm(n)) do if isprime(mm(n)[j]) = true then t := t+1 else end if end do: for j to nops(mm(ithprime(n))) do if isprime(mm(ithprime(n))[j]) = true then s := s+1 else end if end do: if isprime(n) = true and t = s then n else end if end proc: seq(a(n), n = 1 .. 1300); # Emeric Deutsch, Jan 28 2009

Corrected (added 761, 857; deleted 977) and extended by Emeric Deutsch, Jan 28 2009

Edited by Jon E. Schoenfield, Jan 20 2019

A156343 Primes with equal number of prime and nonprime digits.

Original entry on oeis.org

13, 17, 29, 31, 43, 47, 59, 67, 71, 79, 83, 97, 1033, 1123, 1153, 1213, 1217, 1229, 1231, 1259, 1279, 1283, 1297, 1303, 1307, 1321, 1367, 1423, 1427, 1433, 1453, 1531, 1543, 1559, 1567, 1571, 1579, 1583, 1597, 1627, 1637, 1657, 1721, 1747, 1759, 1783, 1787

Offset: 1

Juri-Stepan Gerasimov, Feb 08 2009

Prime digits are 2, 3, 5 or 7. Nonprime digits are 0, 1, 4, 6, 8 or 9.

Complement of (A154385 U A154386). [Juri-Stepan Gerasimov, Nov 29 2009]

			13 (1=nonprime, 3=prime) = a(1).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A000040, A154385, A154386, A371352.

A109066c := proc(n) nops(convert(n,base,10))-A109066(n) ; end:
A109066 := proc(n) local dgs,a ; dgs := convert(n,base,10) ; a := 0 ; for i in dgs do if isprime(i) then a := a+1 ; fi; od: a ; end:
for i from 1 to 400 do p := ithprime(i) ; if A109066(p) = A109066c(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Feb 09 2009

Select[Prime[Range[5,300]],Length[Select[IntegerDigits[#],PrimeQ]]==Length[ Select[ IntegerDigits[ #],!PrimeQ[ #]&]]&] (* Harvey P. Dale, Dec 15 2022 *)

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s, counts = str(n), [0, 0]
    for c in s: counts[int(c in "2357")] += 1
    return counts[0] == counts[1]
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Apr 23 2024

A155552 Primes p such that (number of prime digits of p) > (number of prime digits of prime(p)).

Original entry on oeis.org

5, 13, 23, 29, 43, 53, 79, 127, 157, 167, 173, 223, 227, 229, 233, 239, 257, 263, 271, 277, 283, 293, 313, 317, 337, 353, 373, 379, 433, 523, 557, 563, 577, 647, 757, 773, 797, 839, 853, 859, 863, 887, 977, 1103, 1117, 1123, 1153, 1171, 1187, 1193, 1217

Offset: 1

Juri-Stepan Gerasimov, Jan 24 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A109066, A155555.

Select[Prime[Range[200]],Count[IntegerDigits[#],?PrimeQ]> Count[ IntegerDigits[ Prime[#]],?PrimeQ]&] (* Harvey P. Dale, May 08 2012 *)

Corrected (337 inserted, 761 removed etc.) by R. J. Mathar, May 15 2010

A155555 Primes p such that (number of prime digits of p) < (number of prime digits of prime(p)).

Original entry on oeis.org

11, 19, 31, 41, 59, 61, 67, 71, 101, 103, 107, 109, 131, 137, 139, 149, 151, 181, 191, 199, 211, 241, 269, 281, 307, 311, 331, 349, 367, 389, 401, 409, 419, 439, 449, 457, 461, 463, 467, 479, 491, 499, 503, 509, 521, 569, 587, 599, 601, 613, 619, 641, 643

Offset: 1

Juri-Stepan Gerasimov, Jan 24 2009

Prime digits are 2, 3, 5, 7.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A000040, A109066, A155552.

dp(n)=d=digits(n);c=0;for(i=1,#d,if(isprime(d[i]),c+=1));c
forprime(p=1,500,if(dp(p)Derek Orr, Feb 28 2017

Corrected (87 removed) by R. J. Mathar, May 15 2010

A154762 Primes with the same number of decimal digits that are {1, 9} as {2, 3, 5, 7}.

Original entry on oeis.org

13, 17, 29, 31, 59, 71, 79, 97, 103, 107, 163, 167, 241, 269, 281, 349, 389, 421, 431, 439, 479, 509, 541, 569, 613, 617, 631, 659, 701, 709, 761, 769, 821, 829, 839, 859, 907, 947, 967, 983, 1063, 1087, 1123, 1153, 1213, 1217, 1229, 1231, 1259, 1279, 1297

Offset: 1

Juri-Stepan Gerasimov, Jan 15 2009

Primes with the same number of prime decimal digits as odd nonprime digits.

digs1or9 := proc(n) local a,c; a := 0 ; for c in convert(n,base,10) do if c in {1,9} then a := a+1 ; end if; end do; a; end proc: A109066 := proc(n) local a,d; p := ithprime(n) ; a := 0 ; for d in convert(p,base,10) do if isprime(d) then a := a+1 ; end if; end do; a; end proc: for n from 1 to 400 do p := ithprime(n) ; if digs1or9(p) = A109066(n) then printf("%d,",p) ; fi; od: # R. J. Mathar, Oct 22 2009

ddQ[n_]:=Module[{idn=IntegerDigits[n]},Count[idn,1|9]==Count[idn, 2|3|5|7]]; Select[Prime[Range[250]],ddQ] (* Harvey P. Dale, Nov 29 2011 *)

cp's OEIS Frontend

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

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A034844 Primes with only nonprime decimal digits.

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A033274 Primes that do not contain any other prime as a proper substring.

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A193238 Number of prime digits in decimal representation of n.

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A179336 Primes containing at least one prime digit in base 10.

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A155548 Primes p such that p and the p-th prime have the same number of prime digits.

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A156343 Primes with equal number of prime and nonprime digits.

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