A038603 -id:A038603 - cp's OEIS frontend

A038613 Primes not containing the digit '5'.

2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 263, 269, 271, 277, 281, 283, 293, 307, 311

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Subsequence of primes of A052413. - Michel Marcus, Feb 22 2015

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

			From _M. F. Hasler_, Jan 14 2020: (Start)
After a(85) = 499, the next prime without digit 5 is a(86) = 601.
After a(3734) = 49999, the next term is a(3735) = 60013.
After a(27273) = 499979, the next term is 600011.
After a(206276) = 4999999, the next term is 6000011. (End)

Jason Bard, Table of n, a(n) for n = 1..10000
M. F. Hasler, Numbers avoiding certain digits, OEIS wiki, Jan 12 2020.
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

Intersection of A000040 (primes) and A052413 (numbers with no digit 5).

Primes having no digit d = 0..9 are A038618, A038603, A038604, A038611, A038612, this sequence, A038614, A038615, A038616, and A038617, respectively.

[ p: p in PrimesUpTo(400) | not 5 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 5] == 0 &] (* Vincenzo Librandi, Aug 08 2011 *)

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 5), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

(A038613_upto(n)=select( is_A052413, primes([1, n])))(350) \\ see A052413
next_A038613(n)={until(isprime(n), n=next_A052413(nextprime(n+1)-1)); n}
( {A038613_vec(n, M=1)=M--;vector(n, i, M=next_A038613(M))} )(20, 1000) \\ Compute n terms >= M. See also the OEIS wiki page. - M. F. Hasler, Jan 14 2020

a(n) ≍ n^(log 10/log 9) log n. - Charles R Greathouse IV, Aug 03 2023

Offset corrected by Arkadiusz Wesolowski, Aug 07 2011

A038616 Primes not containing the digit '8'.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 293, 307

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Subsequence of primes of A052421. - Michel Marcus, Feb 22 2015

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

Indranil Ghosh, Table of n, a(n) for n = 1..50000
M. F. Hasler, Numbers avoiding certain digits, OEIS Wiki, Jan 12 2020.
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

Intersection of A000040 (primes) and A052421 (numbers with no 8).

Primes having no digit d = 0..9 are A038618, A038603, A038604, A038611, A038612, A038613, A038614, A038615, this sequence, and A038617, respectively.

[ p: p in PrimesUpTo(400) | not 8 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 8] == 0 &] (* Harvey P. Dale, Jan 24 2011 *)

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 8), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

next_A038616(n)=until((n=nextprime(n+1))==(n=next_A052421(n-1)), ); n \\ M. F. Hasler, Jan 14 2020

a(n) ~ n^(log 10/log 9) * log(n). - Charles R Greathouse IV, Aug 03 2023

Offset corrected by Arkadiusz Wesolowski, Aug 07 2011

A208270 Primes containing a digit 1.

Original entry on oeis.org

11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 241, 251, 271, 281, 311, 313, 317, 331, 401, 419, 421, 431, 461, 491, 521, 541, 571, 601, 613, 617, 619, 631, 641, 661

Offset: 1

Jaroslav Krizek, Mar 04 2012

Subsequence of A011531, A062634, A092911 and A092912.
Supersequence of A106101, A045707 and A030430.
Complement of A208271 with respect to A011531.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of n

Cf. A208271 (nonprimes containing a digit 1), A011531 (numbers containing a digit 1).

Complement of A038603 in A000040. - M. F. Hasler, Mar 05 2012

[p: p in PrimesUpTo(400) | 1 in Intseq(p)]; // Vincenzo Librandi, Apr 29 2019

Select[Prime[Range[124]], MemberQ[IntegerDigits[#], 1] &](* Jayanta Basu, Apr 01 2013 *)
Select[Prime[Range[200]],DigitCount[#,10,1]>0&] (* Harvey P. Dale, Dec 15 2020 *)

forprime(p=2,1e3,s=vecsort(eval(Vec(Str(p))),,8);if(s[1]==1||(s[1]==0&&s[2]==1),print1(p", "))) \\ Charles R Greathouse IV, Mar 04 2012

is_A208270(n)=isprime(n)&setsearch(Set(Vec(Str(n))),1) \\ M. F. Hasler, Mar 05 2012

a(n) ~ n log n since the sequence contains almost all primes. - Charles R Greathouse IV, Mar 04 2012

A106116 Primes without {0, 1} as digits.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 223, 227, 229, 233, 239, 257, 263, 269, 277, 283, 293, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 433, 439, 443, 449, 457, 463, 467, 479, 487, 499, 523, 547, 557, 563, 569, 577, 587, 593

Offset: 1

Zak Seidov, May 07 2005

Indranil Ghosh, Table of n, a(n) for n = 1..50000
Chris C. Caldwell, Yarborough prime
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

Intersection of A038603 and A038618.

Cf. A000040, A106745.

Select[Prime[Range[100]], Min[IntegerDigits[ # ]]>1&]

is(p)=vecsort(digits(p),,8)[1]>1 && isprime(p) \\ Charles R Greathouse IV, Jan 02 2013

a(n) >> n^k with k = log 10/log 8 = 1.107.... - Charles R Greathouse IV, Jan 02 2013

Terms > 523 added by Jonathan Vos Post, Feb 10 2010

A076805 Triskaidekaphobic or 13-free primes: primes that do not contain the number 13.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 127, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307

Offset: 1

Cino Hilliard, Nov 18 2002

			The PARI program will mask out a sequence containing k or mask in a sequence containing k. The program is limited to primes < 400000000.
The PARI program will generate the following for input as shown: kprimes(2,100,7,0) = 2 3 5 11 13 19 23 29 31 41 43 53 59 61 83 89; kprimes(2,1000,13,1) = 13 113 131 137 139 313 613; kprimes(300000,4000000,314159,1) = 314159 3314159 5314159

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triskaidekaphobia
Wikipedia, Triskaidekaphobia

A generalization of the examples A038603, A038615 etc. for k = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 by Vasiliy Danilov.
Cf. A038615 A038603.
Complement of A166573 with respect to A000040; cf. A011760

import Data.List (isInfixOf)
a076805 n = a076805_list !! (n-1)
a076805_list = filter (not . ("13" `isInfixOf`) . show) a000040_list
-- Reinhard Zumkeller, Nov 09 2011

Select[Prime[Range[90]],!MemberQ[Partition[IntegerDigits[#],2,1],{1,3}]&] (* Harvey P. Dale, Mar 25 2012 *)
Select[Prime[Range[100]],SequenceCount[IntegerDigits[#],{1,3}]==0&] (* Harvey P. Dale, Aug 11 2023 *)

/* k primes - kprimes.gp. Primes containing or not containing digit k=1,2,3...9,10,11... PARI does not have a good string manipulation capability. This program circumvents that by using the % modulo and floor operators. Also commented out is a log implementation which is slower than string apps. The program either masks out prime numbers k or masks them in.*/

log10(z) = if(z>0,floor(log(z)/log(10))+1,1);\\integer function for log(z) base 10 + 1
{ kprimes(n1,n2,k,t) = \\n1,n2=range,k=mask,t=0 mask out t=1 mask in
ct=0; pct=0; forprime(p=n1,n2,x=p; f=0;\\x=temp variable to diminish p
ln = length(Str(p));\\get length of the prime p using strings
lk = length(Str(k));\\get length of mask integer k using strings
ln = log10(p);\\get length of the prime p using logs
lk = log10(k);\\get length of mask integer k using strings
r = 10^lk; \\set the remainder length = length of k
for(j=1,ln-lk+1, \\permute through the digits
d = x % r;\\get lk digits
if(d==k,f=1; break);\\break for loop if match and set flag
x = floor(x/10);\\diminish x for next test for k
); if(f==t,print1(p" "); ct+=1);\\if no k string of digits,print
); print(); print(ct); }

hasNo13(n)=n=digits(n); for(i=2,#n, if(n[i]==3&&n[i-1]==1, return(0))); 1
select(hasNo13, primes(10^4)) \\ Charles R Greathouse IV, Dec 02 2013

is_A076805(n,t=13)=!until(t>n\=10,t==n%100&&return) \\ M. F. Hasler, Dec 02 2013

a(n) >> n^1.0044, where the exponent is log(r)/log(10) with r the larger root of x^2 - 10x + 1. [Charles R Greathouse IV, Nov 09 2011]

kfreep(n, k) = true if for primes p over the range n and integer k, p mod 10^(floor(log_10(k))+1) <> k for p = p/10 mod 10^(floor(log_10(k))+1) <> k over the range floor(log_10(k))+1. This is a mathematical definition of the recurrence. In practice it is convenient to use string operations. E.g., If Not Instr(Str(p), Str(k)) then true or k is not a substring of p so list p.

Original PARI code restored by M. F. Hasler, Dec 02 2013

A132080 Numbers having in decimal representation no common digits with all their proper divisors.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 23, 27, 29, 34, 37, 38, 43, 46, 47, 49, 53, 54, 56, 57, 58, 59, 67, 68, 69, 73, 76, 78, 79, 83, 86, 87, 89, 97, 223, 227, 229, 233, 239, 247, 249, 257, 259, 263, 267, 269, 277, 283, 289, 293, 307, 323, 329, 334, 337, 347, 349, 353, 356, 358, 359

Offset: 1

Reinhard Zumkeller, Aug 09 2007

A038603 is a subsequence.

			Proper divisors of a(20)=54: {1, 2, 3, 6, 9, 18, 27}
with set of digits {1,2,3,6,7,8,9} containing neither 4 nor 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Proper Divisor

Subsequence of A052383.

import Data.List (intersect)
a132080 n = a132080_list !! (n-1)
a132080_list = [x | x <- [2..], all null $
                    map (show x `intersect`) $ map show $ a027751_row x]
-- Reinhard Zumkeller, Oct 06 2012

Select[Range[2,359],ContainsNone[IntegerDigits[#],IntegerDigits/@Drop[Divisors[#],-1]//Flatten]&] (* James C. McMahon, Mar 03 2025 *)

is(n)=my(D=Set(digits(n))); fordiv(n,d, if(#setintersect(D, Set(digits(d))), return(d==n&&n>1))) \\ Charles R Greathouse IV, Dec 01 2013

A224319 Primes without "1" as a digit that remain prime when any single digit is replaced with "1".

Original entry on oeis.org

37, 43, 47, 67, 73, 79, 337, 409, 439, 499, 607, 709, 3637, 3709, 4877, 6997, 7487, 9433, 76963, 334777

Offset: 1

Arkadiusz Wesolowski, Apr 03 2013

No more terms < 10^13.

Carlos Rivera, Puzzle 591

Cf. A224320-A224322. Subsequence of A038603.

lst = {}; n = 1; Do[If[PrimeQ[p], i = IntegerDigits[p]; If[FreeQ[i, n], t = 0; s = IntegerLength[p]; Do[If[PrimeQ@FromDigits@Insert[Drop[i, {d}], n, d], t++, Break[]], {d, s}]; If[t == s, AppendTo[lst, p]]]], {p, 334777}]; lst

A084368 Numbers k such that prime(k) does not contain the digit 1.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 12, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 48, 49, 50, 51, 52, 55, 56, 57, 59, 61, 62, 63, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 84, 85, 86, 87, 88, 90, 91, 92, 93, 95, 96, 97, 99, 101, 102, 103, 104, 106, 107, 108, 109, 111, 117, 118, 119, 120

Offset: 1

Zak Seidov, Jun 23 2003

			99 is a term because prime(99) = 523 is unit-free.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000040, A007498, A038603.

Select[ Range[120], Count[ IntegerDigits[ Prime[ # ]], 1] == 0 & ]
Select[Range[120],DigitCount[Prime[#],10,1]==0&] (* Harvey P. Dale, Jun 20 2023 *)

Edited and extended by Robert G. Wilson v, Jun 24 2003

A104421 Numbers n such that n, prime(n), prime(n)+n, prime(n)-n and prime(n)*n all numbers without the digit 1.

Original entry on oeis.org

74, 75, 80, 86, 87, 95, 96, 350, 352, 354, 355, 357, 360, 364, 376, 536, 557, 564, 583, 584, 590, 592, 593, 594, 596, 599, 600, 623, 635, 639, 656, 659, 660, 665, 667, 674, 677, 678, 699, 700, 703, 706, 707, 724, 728, 734, 744, 750, 759, 762, 765, 766, 770

Offset: 1

Zak Seidov, Mar 07 2005

The graph is of quasi-piecewise linear character.
Any other reasonable function(s) of p and m not having digit 1?
Subsequence of A084368: a(1) = 75 = A084368(36), a(100) = 2744 = A084368(898). - Zak Seidov, Dec 04 2013

			For n = 74, p = prime(74) = 373, p + n = 447, p - n = 299, p*n = 27602.
For n = 256709, p = prime(256709) = 3599737, p + n = 3856446, p - n = 3343028, p*n = 924084885533.

Zak Seidov, Table of n, a(n) for n = 1..3000

Cf. A038603, A084368, A104419-A104428.

id[x_]:=IntegerDigits[x]; pr[i_]:=Prime[i]; ra=Range[3000]; A104421=Select[ra, Position[Union[id[ # ], id[pr[ # ]], id[pr[ # ]+# ], id[pr[ # ]-# ], id[pr[ # ]*# ]], 1]=={}&]
prQ[n_]:=With[{p=Prime[n]},AllTrue[IntegerDigits/@{n,p,p+n,p-n,p*n},FreeQ[#,1]&]]; Select[Range[1000],prQ] (* Harvey P. Dale, Aug 29 2025 *)

A111488 Primes having only {0, 1, 3, 6} as digits.

Original entry on oeis.org

3, 11, 13, 31, 61, 101, 103, 113, 131, 163, 311, 313, 331, 601, 613, 631, 661, 1013, 1031, 1033, 1061, 1063, 1103, 1163, 1301, 1303, 1361, 1601, 1613, 1663, 3001, 3011, 3061, 3163, 3301, 3313, 3331, 3361, 3613, 3631, 6011, 6101, 6113, 6131, 6133, 6163

Offset: 1

Jonathan Vos Post, Nov 15 2005

Includes all repunit primes (A004022). Conjecture: an infinite sequence. Note twin primes: (11, 13), (101, 103), (311, 313), (1031, 1033), (1061, 1063), (1301, 1303), (6131, 6133), (10301, 10303), (10331, 10333), (13001, 13003).
In other words, primes with digits in the set {0,1,3,6}. - M. F. Hasler, Jul 25 2015
The number of 1's in the representation must be either 1 or 2 (mod 3), because otherwise the number would be divisible by 3 (and therefore composite). The only exception is the 3 itself. This excludes basically members of A038603. - R. J. Mathar, Jul 25 2015

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

Cf. A000040, A000217, A004022, A038603.

Cf. also A020450 - A020472, A036953, A260044, A260267 - A260271, A199325 - A199329, A061247, A199340 - A199349.

f:= proc(x) local L,p;
  L:= subs([3=6,2=3],convert(x,base,4));
  p:= add(L[i]*10^(i-1),i=1..nops(L));
  if isprime(p) then p fi
end proc:
map(f, [$1..4^4]); # Robert Israel, Dec 18 2018

Select[Prime@ Range@ 1000, SubsetQ[{0, 1, 3, 6}, IntegerDigits@ #] &] (* Michael De Vlieger, Jul 25 2015 *)

A111488={(n, show=0, L=[0,1,3,6])->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)))} \\ M. F. Hasler, Jul 25 2015

Corrected by Ray Chandler, Nov 19 2005

Name changed by Sean A. Irvine, Jul 21 2025

cp's OEIS Frontend

A038613 Primes not containing the digit '5'.

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Magma

Mathematica

PARI

PARI

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A038616 Primes not containing the digit '8'.

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Magma

Mathematica

PARI

PARI

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A208270 Primes containing a digit 1.

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Magma

Mathematica

PARI

PARI

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A106116 Primes without {0, 1} as digits.

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Mathematica

PARI

Formula

Extensions

A076805 Triskaidekaphobic or 13-free primes: primes that do not contain the number 13.

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Haskell

Mathematica

PARI

PARI

PARI

PARI

Formula

Extensions

A132080 Numbers having in decimal representation no common digits with all their proper divisors.

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Haskell