A030096 -id:A030096 - cp's OEIS frontend

A329761 Primes having only {1, 3, 9} as digits.

3, 11, 13, 19, 31, 113, 131, 139, 191, 193, 199, 311, 313, 331, 911, 919, 991, 1193, 1319, 1399, 1913, 1931, 1933, 1993, 1999, 3119, 3191, 3313, 3319, 3331, 3391, 3911, 3919, 3931, 9133, 9199, 9311, 9319, 9391, 9931, 11113, 11119, 11131, 11311, 11393, 11399

Offset: 1

Alois P. Heinz, Nov 20 2019

Original name was: Primes whose product of decimal digits is a power of 3.

Primes whose digit set is a subset of {1,3,9}.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Marianne Freiberger, Primes without 7s.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)
Index to entries for primes with digits in a given set

Subsequence of A030096.

Cf. A000040, A000244, A007954, A020457, A174813.

[p: p in PrimesUpTo(12000) | Set(Intseq(p)) subset [1,3,9]]; // Vincenzo Librandi, Jan 02 2019

Select[Prime[Range[1500]],IntegerQ[Log[3,Times@@IntegerDigits[#]]]&] (* or *) Table[Select[FromDigits/@Tuples[{1,3,9},n],PrimeQ],{n,5}]// Flatten (* Harvey P. Dale, Dec 31 2019 *)

{ A000040 } intersect { A174813 }.

a(n) in { A000040 } and A007954(a(n)) in { A000244 }.

Name changed by Sean A. Irvine, Jul 20 2025

A104638 Number of odd digits in n-th prime.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Mar 18 2005

The only zero is the first term. Sequence is unbounded. - Zak Seidov, Jan 12 2016
From Robert Israel, Jan 12 2016: (Start)
For any N, the asymptotic density of terms >= N is 1.
On the other hand, a(n) = 2 if prime(n) is in A159352, which is conjectured to be infinite.
Record values: a(2) = 1, a(5) = 2, a(30) = 3, a(187) = 4, a(1346) = 5, a(10545) = 6, a(86538) = 7, a(733410) = 8.
(End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A104637, A159352.

Cf. A154764 (1 odd digit), A155071 (2 odd digits), A030096 (all digits odd).

seq(nops(select(type, convert(ithprime(i),base,10),odd)),i=1..100); # Robert Israel, Jan 12 2016
# alternative
A104638 := proc(n)
    local a,dgs,d ;
    ithprime(n) ;
    dgs := convert(%,base,10) ;
    a := 0 ;
    for d in dgs do
        a := a+modp(d,2) ;
    end do:
    a ;
end proc:
seq(A104638(n),n=1..40) ; # R. J. Mathar, Jul 13 2025

Table[Count[IntegerDigits[Prime[n]],?OddQ],{n,100}] (* _Harvey P. Dale, Jan 22 2012 *)
Table[Total[Mod[IntegerDigits[Prime[n]], 2]], {n, 100}] (* Vincenzo Librandi, Jan 13 2016 *)

a(n)=vecsum(digits(prime(n)%2)) \\ Zak Seidov, Jan 12 2016

a(n) = A196564(A000040(n)). - Michel Marcus, Oct 05 2013

A108382 Primes p such that p's set of distinct digits is {1,3,7}.

Original entry on oeis.org

137, 173, 317, 1373, 1733, 3137, 3371, 7331, 11173, 11317, 11731, 13171, 13177, 13337, 13711, 17137, 17317, 17333, 17377, 17713, 17737, 31177, 31337, 31771, 33317, 33713, 37117, 37171, 37313, 37717, 71317, 71333, 71713, 73133, 73331

Offset: 1

Rick L. Shepherd, Jun 01 2005

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A108383 ({1, 3, 9}), A108384 ({1, 7, 9}), A108385 ({3, 7, 9}), A108386 ({1, 3, 7, 9}), A030096 (Primes whose digits are all odd).

S1[1] := {1}: S3[1]:= {3}: S7[1]:= {7}:
S13[1]:= {}: S17[1]:= {}: S37[1]:={}:
S137[1]:= {}:
for n from 2 to 5 do
  S1[n]:= map(t -> 10*t+1, S1[n-1]);
  S3[n]:= map(t -> 10*t+3, S3[n-1]);
  S7[n]:= map(t -> 10*t+7, S7[n-1]);
  S13[n]:= map(t -> 10*t+1, S13[n-1] union S3[n-1]) union
           map(t -> 10*t+3, S13[n-1] union S1[n-1]);
  S17[n]:= map(t -> 10*t+1, S17[n-1] union S7[n-1]) union
           map(t -> 10*t+7, S17[n-1] union S1[n-1]);
  S37[n]:= map(t -> 10*t+3, S37[n-1] union S7[n-1]) union
           map(t -> 10*t+7, S37[n-1] union S3[n-1]);
  S137[n]:= map(t -> 10*t+1, S137[n-1] union S37[n-1]) union
            map(t -> 10*t+3, S137[n-1] union S17[n-1]) union
            map(t -> 10*t+7, S137[n-1] union S13[n-1]);
od:
sort(convert(`union`(seq(select(isprime,S137[n]),n=3..5)),list)); # Robert Israel, Jan 16 2019

Select[Prime[Range[7300]],Union[IntegerDigits[#]]=={1,3,7}&] (* Harvey P. Dale, Jun 11 2013 *)

A108383 Primes p such that p's set of distinct digits is {1,3,9}.

Original entry on oeis.org

139, 193, 1193, 1319, 1399, 1913, 1931, 1933, 1993, 3119, 3191, 3319, 3391, 3911, 3919, 3931, 9133, 9311, 9319, 9391, 9931, 11393, 11399, 11933, 11939, 13339, 13399, 13913, 13931, 13933, 13999, 19139, 19319, 19333, 19391, 19913, 19993, 31139

Offset: 1

Rick L. Shepherd, Jun 01 2005

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

Cf. A108382 ({1, 3, 7}), A108384 ({1, 7, 9}), A108385 ({3, 7, 9}), A108386 ({1, 3, 7, 9}), A030096 (Primes whose digits are all odd).

Select[(Table[FromDigits/@Tuples[{1,3,9},n],{n,3,5}]//Flatten),PrimeQ[#] && Min[ DigitCount[#,10,{1,3,9}]]>0&] (* Harvey P. Dale, Apr 09 2017 *)

A108384 Primes p such that p's set of distinct digits is {1,7,9}.

Original entry on oeis.org

179, 197, 719, 971, 1979, 1997, 7919, 9719, 9791, 11197, 11719, 11779, 11971, 17191, 17791, 17911, 17971, 17977, 19717, 19777, 19979, 19997, 71119, 71191, 71719, 71917, 71971, 71999, 77191, 77719, 79111, 91711, 91771, 91997, 97117, 97171

Offset: 1

Rick L. Shepherd, Jun 01 2005

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A108382 ({1, 3, 7}), A108383 ({1, 3, 9}), A108385 ({3, 7, 9}), A108386 ({1, 3, 7, 9}), A030096 (Primes whose digits are all odd).

Flatten[Table[Select[FromDigits/@Select[Tuples[{1,7,9},n],Union[#]=={1,7,9}&],PrimeQ],{n,3,5}]] (* Harvey P. Dale, Feb 15 2016 *)

lista(nn) = forprime(p=179, nn, if(vecsort(digits(p), , 8)==[1, 7, 9], print1(p, ", "))) \\ Iain Fox, Oct 25 2017

A108385 Primes p such that p's set of distinct digits is {3,7,9}.

Original entry on oeis.org

379, 397, 739, 937, 3739, 3779, 3793, 3797, 7393, 7793, 7933, 7937, 7993, 9337, 9377, 9397, 9733, 9739, 9973, 33739, 33797, 33937, 33997, 37339, 37379, 37397, 37799, 37993, 37997, 39373, 39397, 39733, 39779, 39799, 39937, 39979, 73379, 73939

Offset: 1

Rick L. Shepherd, Jun 01 2005

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

Cf. A108382 ({1, 3, 7}), A108383 ({1, 3, 9}), A108384 ({1, 7, 9}), A108386 ({1, 3, 7, 9}), A030096 (Primes whose digits are all odd).

Table[Select[FromDigits/@Select[Tuples[{3,7,9},n],SubsetQ[#,{3,7,9}]&], PrimeQ],{n,3,5}]//Flatten (* Harvey P. Dale, Sep 15 2016 *)

A091296 Semiprimes with odd digits.

Original entry on oeis.org

9, 15, 33, 35, 39, 51, 55, 57, 77, 91, 93, 95, 111, 115, 119, 133, 155, 159, 177, 319, 335, 339, 355, 371, 377, 391, 393, 395, 511, 515, 517, 519, 533, 535, 537, 551, 553, 559, 573, 579, 591, 597, 713, 717, 731, 737, 753, 755, 771, 779, 791, 793, 799, 913, 917

Offset: 1

Zak Seidov, Feb 22 2004

Semiprimes with odd digits are more numerous than those with even digits, cf. A108636.

Mia Boudreau, Table of n, a(n) for n = 1..1000

Intersection of A001358 and A014261.

Cf. A030096, A108636.

Select[Range[1000], Plus@@Last/@FactorInteger[ # ]==2&&Union[OddQ/@IntegerDigits[ # ]]=={True}&]
PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 938], PrimeFactorExponentsAdded[ # ] == 2 && Union[ OddQ /@ IntegerDigits[ # ]] == {True} &] (* Robert G. Wilson v, Feb 25 2004 *)
Select[Range[1000], PrimeOmega[#]==2 && And@@OddQ[IntegerDigits[#]]&] (* Harvey P. Dale, Jul 12 2011 *)

Corrected and extended by Ray Chandler and Robert G. Wilson v, Feb 25 2004

Edited by N. J. A. Sloane, Apr 20 2007

A066640 Numbers such that all divisors have only odd digits.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, 35, 37, 39, 51, 53, 55, 57, 59, 71, 73, 77, 79, 91, 93, 95, 97, 99, 111, 113, 117, 119, 131, 133, 137, 139, 151, 153, 155, 157, 159, 171, 173, 177, 179, 191, 193, 197, 199

Offset: 1

Amarnath Murthy, Dec 28 2001

Is this sequence infinite? - Charles R Greathouse IV, Sep 07 2012

			77 = 11 * 7 belongs to this sequence but 75 does not as 25 (with a 2) divides 75.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A014261. A030096 is a subsequence.

Select[Range[250], And@@OddQ/@Flatten[IntegerDigits/@Divisors[ # ]]&]

f(n)=vecmin(Vec(Vecsmall(Str(n)))%2)
is(n)=fordiv(n,d,if(!f(d),return(0)));1 \\ Charles R Greathouse IV, Sep 07 2012

from itertools import islice, count
from sympy import divisors
def A066640(): return filter(lambda n: all(set(str(m)) <= {'1','3','5','7','9'} for m in divisors(n,generator=True)), count(1,2))
A066640_list = list(islice(A066640(),20)) # Chai Wah Wu, Nov 22 2021

Corrected and extended by Harvey P. Dale, Jan 01 2002

A090155 Primes of the form identical digits preceded by a 7.

Original entry on oeis.org

7, 71, 73, 79, 733, 7333, 79999, 733333, 799999, 71111111, 799999999, 79999999999, 79999999999999999999999999, 79999999999999999999999999999999999999999999999999

Offset: 1

Robert G. Wilson v, Nov 22 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..24

Subsequence of A030096 and of A030291. Apart from the first term, a subsequence of A235154.

Select[ FromDigits /@ Flatten[ Table[ PadRight[{7}, i, # ] & /@ {1, 2, 3, 4, 5, 6, 7, 8, 9}, {i, 2, 50}], 1], PrimeQ[ # ] &]
Select[FromDigits/@Flatten[Table[PadRight[{7},i,#]&/@{1,3,7,9},{i,50}],1],PrimeQ[#]&]//Union (* Harvey P. Dale, Nov 22 2024 *)

A093172 Primes of the form 10^n - 3.

Original entry on oeis.org

7, 97, 997, 99999999999999997

Offset: 1

Rick L. Shepherd, Mar 26 2004

nonn

Primes of the form (9*10^n - 27)/9. - Vincenzo Librandi, Nov 16 2010
Also primes of the form 9*R_n - 2, where R_n is the repunit (A002275) of length n.
The next term has 140 digits.
a(n) = 10^A089675(n) - 3 = 10^(A056662(n) + 1) - 3. - Farideh Firoozbakht, Nov 27 2013

Makoto Kamada, Prime numbers of the form 99...997.
Index entries for primes involving repunits

Subsequence of A020471 and hence of A030096.

Cf. A002275, A056662, A089675.

Do[If[PrimeQ[10^n - 3], Print[10^n - 3]], {n, 100}] (* Farideh Firoozbakht, Nov 27 2013 *)
Select[Table[FromDigits[PadLeft[{7},n,9]],{n,25}],PrimeQ] (* Harvey P. Dale, Dec 12 2020 *)

for(n=1,9, if(isprime(p=10^n-3), print1(p", "))) \\ Charles R Greathouse IV, Dec 13 2024

Name shortened and old name moved to comments by Alex Ratushnyak, Apr 26 2012

cp's OEIS Frontend

A329761 Primes having only {1, 3, 9} as digits.

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A104638 Number of odd digits in n-th prime.

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A108382 Primes p such that p's set of distinct digits is {1,3,7}.

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A108383 Primes p such that p's set of distinct digits is {1,3,9}.

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A108384 Primes p such that p's set of distinct digits is {1,7,9}.

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A108385 Primes p such that p's set of distinct digits is {3,7,9}.

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A091296 Semiprimes with odd digits.

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A066640 Numbers such that all divisors have only odd digits.

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