A061246 -id:A061246 - cp's OEIS frontend

A061247 Primes having only {0, 1, 8} as digits.

11, 101, 181, 811, 881, 1181, 1801, 1811, 8011, 8081, 8101, 8111, 10111, 10181, 11801, 18181, 80111, 81001, 81101, 81181, 88001, 88801, 88811, 100801, 100811, 101081, 101111, 108011, 108881, 110881, 118081, 118801, 180001, 180181, 180811

Offset: 1

Amarnath Murthy, Apr 23 2001

The intersection with A007500 is listed in A199328. - M. F. Hasler, Nov 05 2011

			a(6) = 1801, 1801 is a prime and consists of only 1, 8 and 0.

Robert Israel, Table of n, a(n) for n = 1..17482

Cf. A061246, A020449-A020472, A199325-A199329.

[NthPrime(n): n in [1..2*10^4] | forall{d: d in Intseq(NthPrime(n)) | d in [0, 1, 8]}]; // Vincenzo Librandi, May 15 2019

N:= 1000: # to get the first N entries
count:= 0:
allowed:= {0,1,8}:
nallowed:= nops(allowed):
subst:= seq(i=allowed[i+1],i=0..nallowed-1);
for d from 1 while count < N do
  for x1 from 1 to nallowed-1 while count < N do
    for t from 0 to nallowed^d-1  while count < N do
      L:= subs(subst,convert(x1*nallowed^d+t,base,nallowed));
      X:= add(L[i]*10^(i-1),i=1..d+1);
      if isprime(X) then
          count:= count+1;
          A[count]:= X;
      fi
od od od:
seq(A[n],n=1..N); # Robert Israel, Apr 20 2014

Select[Prime[Range[50000]],Length[Union[{0,1,8},IntegerDigits[ # ]]] == 3&] (* Stefan Steinerberger, Jun 10 2007 *)
Select[FromDigits/@Tuples[{0,1,8},6],PrimeQ] (* Harvey P. Dale, Jan 12 2016 *)

a(n=50, L=[0, 1, 8], show=0)={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, Nov 05 2011

Corrected and extended by Stefan Steinerberger, Jun 10 2007

A051416 Primes whose digits are composite; primes having only {4, 6, 8, 9} as digits.

Original entry on oeis.org

89, 449, 499, 4649, 4889, 4969, 4999, 6449, 6469, 6689, 6869, 6899, 6949, 8669, 8689, 8699, 8849, 8969, 8999, 9649, 9689, 9949, 44449, 44699, 46489, 46499, 46649, 46889, 48449, 48649, 48869, 48889, 48989, 49499, 49669, 49999, 64489, 64499, 64849, 64969, 66449

Offset: 1

G. L. Honaker, Jr., Jan 17 2000

Primes formed by using only digits 4, 6, 8, 9. Of course, all the terms of this sequence end with 9. - Bernard Schott, Jan 31 2019

			89 is the smallest composite-digit prime and also the only composite-digit prime whose digits are distinct. - _Bernard Schott_, Jan 31 2019

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Chris Caldwell, The Prime Glossary, Composite number
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 89
Index to entries for primes with digits in a given set

Cf. A019546 (with prime digits), A030096 (with odd digits), A061246 (with square digits), A061371 (composite numbers with prime digits).

Subsequence of A061372 and of A152313.

Select[Prime@Range[6500], Intersection[IntegerDigits[ # ], {0, 1, 2, 3, 5, 7}] == {} & ] (* Ray Chandler, Mar 04 2007 *)
With[{c = {4, 6, 8, 9}}, Array[Select[Map[FromDigits@ Append[#, 9] &, Tuples[c, {#}]], PrimeQ] &, 4]] // Flatten (* Michael De Vlieger, Feb 02 2019 *)

Extended by Ray Chandler, Mar 04 2007

A066591 Primes which can be expressed as a concatenation of nonnegative squares.

Original entry on oeis.org

11, 19, 41, 101, 109, 149, 181, 191, 199, 251, 401, 409, 419, 449, 491, 499, 641, 811, 911, 919, 941, 991, 1009, 1019, 1049, 1091, 1109, 1181, 1259, 1289, 1361, 1409, 1481, 1499, 1601, 1609, 1619, 1699, 1811, 1901, 1949, 1999, 2251, 2549, 2591, 3691

Offset: 1

Amarnath Murthy, Dec 21 2001

All terms are == {1,9} mod 10. - Zak Seidov, Jul 16 2015

The surprising prime 162536496481 is the concatenation of the 6 double-digit squares in increasing order (see Prime Curios! link). - Bernard Schott, Nov 19 2020

			96181 is a term as it is a concatenation of 961 and 81 both of which are squares. 100169 is a term as it is a concatenation of 100 and 169 in one way and also that of 1, 0, 0, 16 and 9 in another way.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..2103 from Robert Israel)
Chris K. Caldwell and G. L. Honaker, Jr., 1625364981, Prime Curios!

A061246 and A167535 are subsequences. - Zak Seidov, Jul 16 2015

Cf. A000290, A068493.

N:= 10^4: # to get all terms <= N
catn:= proc(x,y) if y=0 then 10*x else x*10^(ilog10(y)+1)+y fi end proc:
Sq:= {seq(i^2,i=0..floor(sqrt(N)))}: Agenda:= Sq: S:= Sq:
while Agenda <> {} do
Agenda:= select(`<=`,{seq(seq(catn(f,g),f=Agenda),g=Sq)},N) minus S;
S:= S union Agenda;
od:
sort(convert(select(isprime,S),list)); # Robert Israel, Jul 16 2015

from sympy import sieve
from itertools import count, islice
def iscat(w, A):
    return False if len(w) < 2 else any(w[:i] in A and (w[i:] in A or iscat(w[i:], A)) for i in range(1, len(w)))
def agen():
    S = {"0"}
    for d in count(2):
        S |= {str(i*i) for i in range(10**(d-2), 10**(d-1))}
        for p in sieve.primerange(10**(d-1), 10**d):
            if iscat(str(p), S):
                yield p
print(list(islice(agen(), 50))) # Michael S. Branicky, Feb 20 2024

Corrected and extended by Christopher Lund (clund(AT)san.rr.com), Apr 11 2002

A323387 Primes whose digits are distinct square digits, i.e., consisting of only digits 0, 1, 4, 9.

Original entry on oeis.org

19, 41, 109, 149, 401, 409, 419, 491, 941, 1049, 1409, 4019, 4091, 9041

Offset: 1

Bernard Schott, Jan 13 2019

There are only fourteen terms in this sequence, which is a finite subsequence of A061246.

			1049 is the smallest prime containing all the square digits exactly once, and 9041 is the largest one.

Chris K. Caldwell and G. L. Honaker, Jr., 9041, Prime Curios!

Subsequence of A061246. Subsequence of A029743.

A107613 Numbers n such that both n and prime(n) consist of square digits (0,1,4,9).

Original entry on oeis.org

94, 1140, 9109, 10404, 10449, 41104, 110000, 110010, 114101, 910140, 910190, 910191, 940111, 940141, 940409, 11049914, 94011001, 94011004, 94019941, 94019990, 94109190, 94109191, 94440910, 190004449, 190011049, 190011090, 190011091, 190011909, 190011999, 190149101

Offset: 1

Zak Seidov, May 17 2005

Corresponding primes are: 491, 9199, 94441, 109441, 109919, 494441, 1441049, 1441199, 1499149, which form a subsequence of A061246 (primes with square digits).

Chai Wah Wu, Table of n, a(n) for n = 1..216

Cf. A061246.

Do[id=Union[IntegerDigits[Prime[n]], IntegerDigits[n]];If[Count[id, 2]+Count[id, 3]+Count[id, 5]+Count[id, 6]+Count[id, 7]+Count[id, 8]==0, Print[n]], {n, 200000}]

a(10)-a(30) from Chai Wah Wu, Jan 30 2019

A158290 Primes in which a digit is square of another digit.

Original entry on oeis.org

11, 19, 29, 31, 41, 43, 47, 59, 61, 71, 79, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181

Offset: 1

Parthasarathy Nambi, Mar 15 2009

Different from A061246 and A158051.

There are at most 5*x^k primes missing from this sequence up to x, where k = log 6/log 10 = 0.7781.... - Charles R Greathouse IV, Nov 29 2022

			181 contains two digits which are square of another digit.

Chris Caldwell, The First 1,000 Primes

Cf. A061246, A158051.

A331346 Primes using all the square digits {0, 1, 4, 9} and no others.

Original entry on oeis.org

1049, 1409, 4019, 4091, 9041, 10499, 10949, 14009, 49019, 49109, 90149, 90401, 94109, 99041, 99401, 100049, 101149, 101419, 101449, 104009, 104119, 104149, 104491, 104911, 104999, 109049, 109141, 109441, 110419, 110491, 111049, 111409, 114901, 140009, 140191, 140419

Offset: 1

K. D. Bajpai, Jan 14 2020

Subsequence of A061246.

			a(1) = 1049 is prime containing all the square digits (0, 1, 4, 9) and no others.
a(2) = 1409 is prime containing all the square digits (0, 1, 4, 9) and no others.

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

Cf. A000040, A061246, A066591, A323387.

[p:p in PrimesUpTo(150000)|Set(Intseq(p)) eq {0,1,4,9}]; // Marius A. Burtea, Jan 14 2020

f:= proc(n) local L,x;
  L:= convert(n,base,4);
  if convert(L,set) <> {0,1,2,3} then return NULL fi;
  L:= subs(2=4,3=9,L);
  x:= add(L[i]*10^(i-1),i=1..nops(L));
  if isprime(x) then x else NULL fi
end proc:
map(f, [$4^3..4^6]); # Robert Israel, Jan 16 2020

Select[FromDigits /@ Tuples[{0, 1, 4, 9}, 6], PrimeQ[#] && Union[IntegerDigits[#]] == {0, 1, 4, 9} &]

A106808 Numbers k such that 4440011 * 10^k - 1 is prime.

Original entry on oeis.org

12, 44, 75, 237, 3186, 4379, 5535, 6363, 13707, 14463

Offset: 1

Jason Earls, May 18 2005

nonn

These are primes with square digits and they have all been certified. No more terms up to 37000. Primality proof for the largest: PFGW Version 20041001.Win_Stable (v1.2 RC1b) [FFT v23.8] Primality testing 4440011*10^14463-1 [N+1, Brillhart-Lehmer-Selfridge] Running N+1 test using discriminant 3, base 1+sqrt(3) 4440011*10^14463-1 is prime! (268.8294s+0.0548s)

R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.

Cf. A061246.

is(n)=ispseudoprime(4440011*10^n-1) \\ Charles R Greathouse IV, Jun 13 2017

cp's OEIS Frontend

A061247 Primes having only {0, 1, 8} as digits.

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A051416 Primes whose digits are composite; primes having only {4, 6, 8, 9} as digits.

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A066591 Primes which can be expressed as a concatenation of nonnegative squares.

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A323387 Primes whose digits are distinct square digits, i.e., consisting of only digits 0, 1, 4, 9.

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A107613 Numbers n such that both n and prime(n) consist of square digits (0,1,4,9).

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A158290 Primes in which a digit is square of another digit.

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A331346 Primes using all the square digits {0, 1, 4, 9} and no others.

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A106808 Numbers k such that 4440011 * 10^k - 1 is prime.

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