A036953 -id:A036953 - cp's OEIS frontend

A090726 Duplicate of A036953.

2, 11, 101, 211, 1021, 1201, 2011, 2111, 2221, 10111, 10211, 12011, 12101, 12211

Offset: 1

dead

A020449 Primes whose greatest digit is 1.

11, 101, 10111, 101111, 1011001, 1100101, 10010101, 10011101, 10100011, 10101101, 10110011, 10111001, 11000111, 11100101, 11110111, 11111101, 100100111, 100111001, 101001001, 101001011, 101100011, 101101111, 101111011, 101111111

Offset: 1

David W. Wilson

Primes which are the sums of distinct powers of 10. - Amarnath Murthy, Nov 19 2002
Subsequence of A007088. - Michel Marcus, Dec 18 2015
These numbers are called Anti-Yarborough prime numbers in the Prime Glossary. - Randy L. Ekl, Jan 19 2019

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
C. K. Caldwell, The Prime Glossary, Yarborough prime.
FactorDB, Concatenation of first 35 terms is prime
Index to entries for primes with digits in a given set

Subsequence of A036953.

Cf. A007088, A036952.

[p: p in PrimesUpTo(101111111) | Set(Intseq(p)) subset [0,1]]; // Vincenzo Librandi, Jul 27 2012

N:= 10: # to get all entries with <= N digits
S:= {}:
for d from 1 to N-1 do
  S:= S union select(isprime,map(`+`,map(convert,combinat[powerset]({seq(10^i,i=0..d-1)}),`+`),10^d));
od:
S; # if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, May 04 2015

Select[FromDigits/@Tuples[{0,1},16],PrimeQ] (* Hans Havermann, May 12 2025 *)

is(n)=isprime(n)&&vecmax(digits(n))==1 \\ Charles R Greathouse IV, Jul 01 2013

from sympy import isprime
A020449_list = [n for n in (int(format(m,'b')) for m in range(1,2**10)) if isprime(n)] # Chai Wah Wu, Dec 17 2015

A004022 Primes of the form (10^k - 1)/9. Also called repunit primes or repdigit primes.

Original entry on oeis.org

11, 1111111111111111111, 11111111111111111111111

Offset: 1

N. J. A. Sloane

The next term corresponds to k = 317 and is too large to include: see A004023.
Also called repunit primes or prime repunits.
Also, primes with digital product = 1.
The number of 1's in these repunits must also be prime. Since the number of 1's in (10^k-1)/9 is k, if k = p*m then (10^(p*m)-1) = (10^p)^m-1 => (10^p-1)/9 = q and q divides (10^k-1). This follows from the identity a^k - b^k = (a-b)*(a^(k-1) + a^(k-2)*b + ... + b^(k-1)). - Cino Hilliard, Dec 23 2008
A subset of A020449, ..., A020457, A036953, ..., cf. link to OEIS index. - M. F. Hasler, Jul 27 2015
The terms in this sequence, except 11 which is not Brazilian, are prime repunits in base ten, so they are Brazilian primes belonging to A085104 and A285017. - Bernard Schott, Apr 08 2017

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, p. 11. Graham, Knuth and Patashnik, Concrete mathematics, Addison-Wesley, 1994; see p. 146, problem 22.
M. Barsanti, R. Dvornicich, M. Forti, T. Franzoni, M. Gobbino, S. Mortola, L. Pernazza and R. Romito, Il Fibonacci N. 8 (included in Il Fibonacci, Unione Matematica Italiana, 2011), 2004, Problem 8.10.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

T. D. Noe, Table of n, a(n) for n = 1..5
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See p. 18.
Makoto Kamada, Factorizations of 11...11 (Repunit).
D. H. Lehmer, On the number (10^23-1)/9, Bull. Amer. Math. Soc. 35 (1929), 349-350.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)
Andy Steward, Prime Generalized Repunits
S. S. Wagstaff, Jr., The Cunningham Project
Index to entries for primes with digits in a given set.

Subsequence of A020449.
A116692 is another version of repunit primes or repdigit primes. - N. J. A. Sloane, Jan 22 2023
See A004023 for the number of 1's.
Cf. A046413.

[a: n in [0..300] | IsPrime(a) where a is (10^n - 1) div 9 ]; // Vincenzo Librandi, Nov 08 2014

lst={}; Do[If[PrimeQ[p = (10^n - 1)/9], AppendTo[lst, p]], {n, 10^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *)
Select[Table[(10^n - 1) / 9, {n, 500}], PrimeQ] (* Vincenzo Librandi, Nov 08 2014 *)
Select[Table[FromDigits[PadRight[{},n,1]],{n,30}],PrimeQ] (* Harvey P. Dale, Apr 07 2018 *)

forprime(x=2,20000,if(ispseudoprime((10^x-1)/9),print1((10^x-1)/9","))) \\ Cino Hilliard, Dec 23 2008

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from (t for t in (int("1"*k) for k in count(1)) if isprime(t))
print(list(islice(agen(), 4))) # Michael S. Branicky, Jun 09 2022

a(n) = A002275(A004023(n)).

Edited by Max Alekseyev, Nov 15 2010

Name expanded by N. J. A. Sloane, Jan 22 2023

A036952 Numbers whose binary expansion is a decimal prime.

Original entry on oeis.org

3, 5, 23, 47, 89, 101, 149, 157, 163, 173, 179, 185, 199, 229, 247, 253, 295, 313, 329, 331, 355, 367, 379, 383, 405, 425, 443, 453, 457, 471, 523, 533, 539, 565, 583, 587, 595, 631, 643, 647, 653, 659, 671, 675, 689, 703, 709, 755, 781, 785, 815, 841, 855

Offset: 1

Patrick De Geest, Jan 04 1999

A100051(f(a(n))) = 1 with f(x) = if x<2 then x else 10*f(floor(x/2)) + x mod 2. - Reinhard Zumkeller, Mar 31 2010

Primes in A007088. - N. J. A. Sloane, Feb 17 2023

			1 = 1_2 is not a prime.
2 = 10_2 is not OK because 10 = 2*5 is not a prime.
3 = 11_2 is OK because 11 is a prime.
4 = 100_2 is not OK because 100 = 4*25 is not a prime.
5 = 101_2 is OK because 101 is a prime.
7 = 111_2 is not OK because 111 = 3*37.
11 = 1011_2 is not OK because 1011 = 3*337.
313 = 100111001_2 is OK because 100111001 is prime.

Cf. A007088, A020449, A036953-A036964.

Union of A156059 and A065720.

A007088 := proc(n)
dgs := convert(n,base,2) ;
add(op(i,dgs)*10^(i-1),i=1..nops(dgs)) ;
end proc:
isA036952 := proc(n)
isprime( A007088(n)) :
end proc:
A036952 := proc(n)
if n =1 then
3;
else
for a from procname(n-1)+1 do
if isA036952(a) then
return a ;
end if;
end do:
end if;
end proc:
seq(A036952(n),n=1..80) ;
# R. J. Mathar, Mar 12 2010
A036952 := proc() if isprime(convert(n,binary)) then RETURN (n); fi; end: seq(A036952(), n=1..1000);  # K. D. Bajpai, Jul 04 2014

f[n_,k_]:=FromDigits[IntegerDigits[n,k]];lst={};Do[If[PrimeQ[f[n,2]],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
NestList[NestWhile[# + 2 &, #, ! PrimeQ[FromDigits[IntegerDigits[#2, 2]]] &, 2] &, 3, 52] (* Jan Mangaldan, Jul 02 2020 *)

is(n)=my(v=binary(n));isprime(sum(i=1,#v,v[i]*10^(#v-i))) \\ Charles R Greathouse IV, Jun 28 2013

Entry revised by R. J. Mathar and N. J. A. Sloane, Mar 12 2010

A199329 Primes having only {0, 1, 9} as digits.

Original entry on oeis.org

11, 19, 101, 109, 191, 199, 911, 919, 991, 1009, 1019, 1091, 1109, 1901, 1999, 9001, 9011, 9091, 9109, 9199, 9901, 10009, 10091, 10099, 10111, 10909, 11119, 11909, 19001, 19009, 19919, 19991, 90001, 90011, 90019, 90191, 90199, 90901, 90911, 91009, 91019, 91099, 91199, 91909, 99109, 99119, 99191, 99901, 99991

Offset: 1

M. F. Hasler, Nov 05 2011

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

Cf. A020449 - A020472, A036953, A260044, A260267 - A260271, A199325 - A199327, A061247, A199340 - A199349.

Select[FromDigits/@Tuples[{0,1,9},5],PrimeQ] (* Harvey P. Dale, Dec 10 2016 *)

A199329(n=50,show=0,L=[0,1,9])={for(d=1,1e9,my(t,u=vector(d,i,10^(d-i))~);forvec(v=vector(d,i,[1+!(L[1]||(i>1&&iM. F. Hasler, Jul 25 2015

A062339 Primes whose sum of digits is 4.

Original entry on oeis.org

13, 31, 103, 211, 1021, 1201, 2011, 3001, 10111, 20011, 20101, 21001, 100003, 102001, 1000003, 1011001, 1020001, 1100101, 2100001, 10010101, 10100011, 20001001, 30000001, 101001001, 200001001, 1000000021, 1000001011, 1000010101, 1000020001, 1000200001, 1002000001, 1010000011

Offset: 1

Amarnath Murthy, Jun 21 2001

Is this sequence (and its brothers A062337, A062341 and A062343) infinite?

10^A049054(m)+3 and 3*10^A056807(m)+1 are subsequences. A107715 (primes containing only digits from set {0,1,2,3}) is a supersequence. Terms not containing the digit 3 are either terms of A020449 (primes that contain digits 0 and 1 only) or of A106100 (primes with maximal digit 2) - and thus terms of these sequences' union A036953 (primes containing only digits from set {0,1,2}). - Rick L. Shepherd, May 23 2005

			3001 is a prime with sum of digits = 4, hence belongs to the sequence.

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..1000 from T. D. Noe)
Amin Witno, Numbers which factor as their digital sum times a prime, International Journal of Open Problems in Computer Science and Mathematics 3:2 (2010), pp. 132-136.

Subsequence of A062338, A107288, and A107715 (primes with digits <= 3).
A159352 is a subsequence.
Cf. A000040 (primes), A052218 (digit sum = 4), A061239 (primes == 4 (mod 9)).
Cf. Primes p with digital sum equal to k: {2, 11 and 101} for k=2; this sequence (k=4), A062341 (k=5), A062337 (k=7), A062343 (k=8), A107579 (k=10), A106754 (k=11), A106755 (k=13), A106756 (k=14), A106757 (k=16), A106758 (k=17), A106759 (k=19), A106760 (k=20), A106761 (k=22), A106762 (k=23), A106763 (k=25), A106764 (k=26), A048517 (k=28), A106766 (k=29), A106767 (k=31), A106768 (k=32), A106769 (k=34), A106770 (k=35), A106771 (k=37), A106772 (k=38), A106773 (k=40), A106774 (k=41), A106775 (k=43), A106776 (k=44), A106777 (k=46), A106778 (k=47), A106779 (k=49), A106780 (k=50), A106781 (k=52), A106782 (k=53), A106783 (k=55), A106784 (k=56), A106785 (k=58), A106786 (k=59), A106787 (k=61), A107617 (k=62), A107618 (k=64), A107619 (k=65), A106807 (k=67), A244918 (k=68), A181321 (k=70).
Cf. A049054 (10^k+3 is prime), A159352 (these primes).
Cf. A056807 (3*10^k+1 is prime), A259866 (these primes).
Cf. A020449 (primes with digits 0 and 1), A036953 (primes with digits <= 2), A106100 (primes with largest digit = 2), A069663, A069664 (smallest resp. largest n-digit prime with minimum digit sum).

[p: p in PrimesUpTo(800000000) | &+Intseq(p) eq 4]; // Vincenzo Librandi, Jul 08 2014

N:= 20: # to get all terms < 10^N
B[1]:= {1}:
B[2]:= {2}:
B[3]:= {3}:
A:= {}:
for d from 2 to N do
   B[4]:= map(t -> 10*t+1,B[3]) union  map(t -> 10*t+3,B[1]);
   B[3]:= map(t -> 10*t, B[3]) union map(t -> 10*t+1,B[2]) union map(t -> 10*t+2,B[1]);
   B[2]:= map(t -> 10*t, B[2]) union map(t -> 10*t+1,B[1]);
   B[1]:= map(t -> 10*t, B[1]);
   A:= A union select(isprime,B[4]);
od:
sort(convert(A,list)); # Robert Israel, Dec 28 2015

Union[FromDigits/@Select[Flatten[Table[Tuples[{0,1,2,3},k],{k,9}],1],PrimeQ[FromDigits[#]]&&Total[#]==4&]] (* Jayanta Basu, May 19 2013 *)
FromDigits/@Select[Tuples[{0,1,2,3},10],Total[#]==4&&PrimeQ[FromDigits[#]]&] (* Harvey P. Dale, Jul 23 2025 *)

for(a=1,20,for(b=0,a,for(c=0,b,if(isprime(k=10^a+10^b+10^c+1),print1(k", "))))) \\ Charles R Greathouse IV, Jul 26 2011

select( {is_A062339(p,s=4)=sumdigits(p)==s&&isprime(p)}, primes([1,10^7])) \\ 2nd optional parameter for similar sequences with other digit sums. M. F. Hasler, Mar 09 2022

A062339_upto_length(L,s=4,a=List(),u=[10^(L-k)|k<-[1..L]])=forvec(d=[[1,L]|i<-[1..s]], isprime(p=vecsum(vecextract(u,d))) && listput(a,p),1); Vecrev(a) \\ M. F. Hasler, Mar 09 2022

Intersection of A052218 (digit sum 4) and A000040 (primes). - M. F. Hasler, Mar 09 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001
More terms from Rick L. Shepherd, May 23 2005
More terms from Lekraj Beedassy, Dec 19 2007

A260266 Primes having only {0, 1, 4} as digits.

Original entry on oeis.org

11, 41, 101, 401, 4001, 4111, 4441, 10111, 10141, 11411, 14011, 14401, 14411, 40111, 41011, 41141, 41411, 44041, 44101, 44111, 100411, 101111, 101141, 101411, 110441, 114001, 114041, 140111, 140401, 140411, 141041, 141101, 400441, 401101, 401411, 404011

Offset: 1

Vincenzo Librandi, Jul 22 2015

A020449 and A020452 are subsequences.

All terms end with a digit "1". - M. F. Hasler, Jul 26 2015

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

Primes that contain only digits among {1,4,k}: this sequence (k=0), A260267 (k=2), A199341 (k=3), A260268 (k=5), A260269 (k=6), A079651 (k=7), A260270 (k=8), A260271 (k=9).
Cf. A020449, A020452.
Cf. A020450 - A020472, A036953, A260044, A260267 - A260271, A199325 - A199329, A061247, A199340 - A199349.

[p: p in PrimesUpTo(5*10^5) | Set(Intseq(p)) subset [1, 4, 0]];

Select[Prime[Range[4 10^4]], Complement[IntegerDigits[#], {1, 4, 0}]=={} &]

A260266(n=50,show=0)={for(d=1,1e9,my(t,u=vector(d,i,10^(d-i))~);forvec(v=vector(d,i,[i==1||i==d,1+(iM. F. Hasler, Jul 25 2015

A199327 Primes having only {0, 1, 7} as digits.

Original entry on oeis.org

7, 11, 17, 71, 101, 107, 701, 1117, 1171, 1777, 7001, 7177, 7717, 10007, 10111, 10177, 10711, 10771, 11071, 11117, 11171, 11177, 11701, 11717, 11777, 17011, 17077, 17107, 17117, 17707, 70001, 70111, 70117, 70177, 70717, 71011, 71171, 71707, 71711, 71777, 77017, 77101, 77171, 77711, 101107, 101111, 101117

Offset: 1

M. F. Hasler, Nov 05 2011

Harvey P. Dale, Table of n, a(n) for n = 1..10000
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).

Cf. A020449 - A020472, A036953, A260044, A260267 - A260271, A199325 - A199329, A061247, A199340 - A199349.

[p: p in PrimesUpTo(80000) | Intseq(p) subset {0,1,7}]; // Vincenzo Librandi, Jan 16 2020

f[i_,nn_]:=Select[Flatten[Table[FromDigits/@(Join[{i},#]&/@Tuples[ {0,1,7}, n]), {n,0,nn}]],PrimeQ]; Union[Join[f[1,6],f[7,6]]] (* Harvey P. Dale, Nov 19 2011 *)
Select[Prime[Range[2 10^4]], Complement[IntegerDigits[#], {0, 1, 7}]=={}&] (* Vincenzo Librandi, Jan 16 2020 *)

a(n,list=0,L=[0,1,7])={for(d=1,1e9,my(t,u=vector(d,i,10^(d-i))~);forvec(v=vector(d,i,[1+!(L[1]||(i>1&&iM. F. Hasler, Jul 26 2015

A260044 Primes having only {0, 1, 3} as digits.

Original entry on oeis.org

3, 11, 13, 31, 101, 103, 113, 131, 311, 313, 331, 1013, 1031, 1033, 1103, 1301, 1303, 3001, 3011, 3301, 3313, 3331, 10103, 10111, 10133, 10301, 10303, 10313, 10331, 10333, 11003, 11113, 11131, 11311, 13001, 13003, 13033, 13103, 13313, 13331, 30011, 30013, 30103, 30113, 30133, 30313, 31013, 31033, 31333, 33013

Offset: 1

M. F. Hasler, Jul 25 2015

A subsequence of A107715 and of A111488.

Number of terms < 10^n: 1, 4, 11, 22, 54, 118, 293, 691, 1837, 4871, 13321, 36042, 98325, 272237, 757080, .... - Robert G. Wilson v, Jul 26 2015

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

Cf. A020449 - A020472, A036953, A260266 - A260271, A199325 - A199329, A061247, A199340 - A199349.

[p: p in PrimesUpTo(10^5) | Set(Intseq(p)) subset [0, 1, 3]]; // Vincenzo Librandi, Jul 26 2015

Select[ FromDigits@# & /@ Tuples[{0, 1, 3}, 5], PrimeQ] (* Robert G. Wilson v, Jul 26 2015 *)
Select[Prime[Range[4 10^3]], Complement[IntegerDigits[#], {0, 1, 3}]=={} &] (* Vincenzo Librandi, Jul 26 2015 *)

A260044={(n,show=0,L=[0,1,3])->my(t);for(d=1,1e9,u=vector(d,i,10^(d-i))~;forvec(v=vector(d,i,[1+(i==1&!L[1]),#L]),ispseudoprime(t=vector(d,i,L[v[i]])*u)||next;show&print1(t",");n--||return(t)))}

A217039 Primes having only {4, 5, 7} as digits.

Original entry on oeis.org

5, 7, 47, 457, 547, 557, 577, 757, 4447, 4457, 4547, 5477, 5557, 7457, 7477, 7547, 7577, 7757, 44777, 45557, 45757, 47777, 54547, 54577, 55457, 55547, 57457, 57557, 74747, 75557, 75577, 77447, 77477, 77557, 77747, 444547, 444557, 445447, 445477, 445747, 447757

Offset: 1

Jonathan Vos Post, Sep 24 2012

These are the primes in A214584. Primes whose numerals are all written (san serif) with at least one right or acute angle.

Cf. A079651, A079652, A214584.

Cf. A036953, A260044, A260267 - A260271, A199325 - A199329, A061247, A199340 - A199349.

[p: p in PrimesUpTo(450000) | Intseq(p) subset [4,5,7]]; // Bruno Berselli, Sep 25 2012

Select[Flatten[Table[FromDigits/@Tuples[{4,5,7},n],{n,6}]],PrimeQ] (* Bruno Berselli, Sep 25 2012 *)

A217039(n=50,show=0,L=[4,5,7])={for(d=1,1e9, my(t, u=vector(d,i,10^(d-i))~); forvec(v=vector(d,i,[if(i==d&&d>1,3/*must end in 7*/,1), #L]), ispseudoprime(t=vecextract(L, v)*u)||next; show&&print1(t", "); n--||return(t)))} \\ Syntax updated for newer PARI versions by M. F. Hasler, Jul 25 2015

A000040 INTERSECTION A214584.

cp's OEIS Frontend

A090726 Duplicate of A036953.

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A020449 Primes whose greatest digit is 1.

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A004022 Primes of the form (10^k - 1)/9. Also called repunit primes or repdigit primes.

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A036952 Numbers whose binary expansion is a decimal prime.

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A199329 Primes having only {0, 1, 9} as digits.

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A062339 Primes whose sum of digits is 4.

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A260266 Primes having only {0, 1, 4} as digits.

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