A036952 -id:A036952 - cp's OEIS frontend

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

A065720 Primes whose binary representation is also the decimal representation of a prime.

Original entry on oeis.org

3, 5, 23, 47, 89, 101, 149, 157, 163, 173, 179, 199, 229, 313, 331, 367, 379, 383, 443, 457, 523, 587, 631, 643, 647, 653, 659, 709, 883, 947, 997, 1009, 1091, 1097, 1163, 1259, 1277, 1283, 1289, 1321, 1483, 1601, 1669, 1693, 1709, 1753, 1877, 2063, 2069, 2099

Offset: 1

Patrick De Geest, Nov 15 2001

In general rebase notation (Marc LeBrun): p2 = (2) [p] (10).
Also: Primes in A036952. - M. F. Hasler, Dec 11 2012
See A089971 for the binary representation of these terms. - M. F. Hasler, Jan 05 2014

			1009{10} = 1111110001{2} is prime, and 1111110001{10} is also prime.
89 is in the sequence because it is a prime. Binary representation of 89 = 1011001, which is also a prime.

Harry J. Smith and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime
Carlos Rivera, Puzzle 280. 3893257, The Prime Puzzles & Problems Connection.

Cf. A065721, A065722, A065723, A065724, A065725, A065726, A065727, A065361.

Cf. A123266, A256621, A256622.

select(t -> isprime(t) and isprime(convert(t,binary)),[seq(2*i+1,i=1..1000)]); # Robert Israel, Jul 08 2014

Select[ Range[1900], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 2]]] & ]
Select[ Prime@ Range@ 330, PrimeQ[ FromDigits[ IntegerDigits[#, 2]]] &] (* Robert G. Wilson v, Oct 09 2014 *)

isok(p) = isprime(p) && isprime(fromdigits(binary(p), 10)); \\ Michel Marcus, Mar 04 2022

from sympy import isprime
def ok(n): return isprime(n) and isprime(int(bin(n)[2:]))
print([k for k in range(2100) if ok(k)]) # Michael S. Branicky, Mar 04 2022

Equals A036952 intersect A000040. - M. F. Hasler, Dec 11 2012

a(48)-a(50) from K. D. Bajpai, Jul 04 2014

A235265 Primes whose base-3 representation also is the base-2 representation of a prime.

Original entry on oeis.org

3, 13, 31, 37, 271, 283, 733, 757, 769, 1009, 1093, 2281, 2467, 2521, 2551, 2917, 3001, 3037, 3163, 3169, 3187, 3271, 6673, 7321, 7573, 9001, 9103, 9733, 19801, 19963, 20011, 20443, 20521, 20533, 20749, 21871, 21961, 22123, 22639, 22717, 27253, 28711, 28759, 29173, 29191, 59077, 61483, 61507, 61561, 65701, 65881

Offset: 1

M. F. Hasler, Jan 05 2014

This sequence and A235383 and A229037 are winners in the contest held at the 2014 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 20 2014
This sequence was motivated by work initiated by V.J. Pohjola's post to the SeqFan list, which led to a clarification of the definition and correction of some errors, in sequences A089971, A089981 and A090707 through A090721. These sequences use "rebasing" (terminology of A065361) from some base b to base 10. Sequences A065720 - A065727 follow the same idea but use rebasing in the other sense, from base 10 to base b. The observation that only (10,b) and (b,10) had been considered so far led to the definition of this and related sequences: In a systematic approach, it seems natural to start with the smallest possible pairs of different bases, (2,3) and (3,2), then (2 <-> 4), (3 <-> 4), (2 <-> 5), etc.
Among the two possibilities using the smallest possible bases, 2 and 3, the present one seems a little bit more interesting, among others because not every base-3 representation is a valid base-2 representation (in contrast to the opposite case). This is also a reason why the present sequence grows much faster than the partner sequence A235266.

			3 = 10_3 and 10_2 = 2 is prime. 13 = 111_3 and 111_2 = 7 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime
Veikko Pohjola, A090709 Decimal primes whose decimal representation in base 6 is also prime, SeqFan list, Jan 03 2014.

Subset of A077717.

Cf. A235266, A065720 and A036952, A065721 - A065727, A235394, A235395, A089971 and A020449, A089981, A090707 - A091924, A235461 - A235482. See M. F. Hasler's OEIS wiki page for further cross-references.

N:= 1000: # to get the first N terms
count:= 0:
for i from 1 while count < N do
   p2:= ithprime(i);
   L:= convert(p2,base,2);
   p3:= add(3^(j-1)*L[j],j=1..nops(L));
   if isprime(p3) then
      count:= count+1;
      A235265[count]:= p3;
   fi
od:
[seq(A235265[i], i=1..N)]; # Robert Israel, May 04 2014

b32pQ[n_]:=Module[{idn3=IntegerDigits[n,3]},Max[idn3]<2&&PrimeQ[ FromDigits[ idn3,2]]]; Select[Prime[Range[7000]],b32pQ] (* Harvey P. Dale, Apr 24 2015 *)

is(p,b=2,c=3)=vecmax(d=digits(p,c))

from sympy import isprime, nextprime
def agen(): # generator of terms
    p = 2
    while True:
        p3 = sum(3**i for i, bi in enumerate(bin(p)[2:][::-1]) if bi=='1')
        if isprime(p3):
            yield p3
        p = nextprime(p)
g = agen()
print([next(g) for n in range(1, 52)]) # Michael S. Branicky, Jan 16 2022

A235482 Primes whose base-5 representation is also the base-9 representation of a prime.

Original entry on oeis.org

2, 3, 7, 11, 17, 19, 37, 41, 61, 67, 71, 97, 109, 131, 139, 149, 151, 157, 167, 191, 197, 211, 251, 269, 281, 337, 349, 367, 401, 409, 439, 449, 457, 467, 487, 491, 499, 521, 557, 569, 607, 619, 631, 647, 661, 739, 761, 769, 821, 829, 887, 907, 941, 947, 967, 1009, 1019, 1031, 1061, 1069, 1087

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

A subsequence of A197636 and of course of A000040 ⊂ A015919.

			41 = 131_5 and 131_9 = 109 are both prime, so 41 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235265, A235266, A235461 - A235481, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395. See the LINK for further cross-references.

Select[Prime@ Range@ 500, PrimeQ@ FromDigits[ IntegerDigits[#, 5], 9] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=9,c=5)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: Code only valid for b > c.

A036953 Primes having only {0, 1, 2} as digits.

Original entry on oeis.org

2, 11, 101, 211, 1021, 1201, 2011, 2111, 2221, 10111, 10211, 12011, 12101, 12211, 20011, 20021, 20101, 20201, 21001, 21011, 21101, 21121, 21211, 21221, 22111, 101021, 101111, 101221, 102001, 102101, 102121, 110221, 111121, 111211, 112111

Offset: 1

Patrick De Geest, Jan 04 1999

Number of n-digit terms d(n) = (1, 1, 2, 5, 16, 34, 76, 194, 543, 1469, 4094, 11017, ...); e.g., there are five 4-digit terms: 1021, 1201, 2011, 2111, 2221, hence d(4) = 5. - Zak Seidov, Jun 30 2013

Also, primes in A007089. - M. F. Hasler, Jul 25 2015

Zak Seidov, Table of n, a(n) for n = 1..10000
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 A107715.
Cf. A036952-A036964.
Cf. A020450 - A020472, A260044, A260267 - A260271, A199325 - A199329, A061247, A199340 - A199349, A217039, A079651.

Select[FromDigits/@Tuples[{0,1,2},6],PrimeQ] (* Harvey P. Dale, Jul 11 2017 *)

lista(n) = {forprime(p=2, n, if (vecmax(digits(p)) <= 2, print1(p, ", ")))} \\ Michel Marcus, Aug 02 2014

A036953={(n,show=0)->for(d=1,1e9,my(u=vector(d,i,10^(d-i))~);forvec(v=vector(d,i,if(i>1,if(iM. F. Hasler, Jul 25 2015

from gmpy2 import digits
from sympy import isprime
[int(digits(n,3)) for n in range(1000) if isprime(int(digits(n,3)))] # Chai Wah Wu, Jul 31 2014

Edited by M. F. Hasler, Jul 25 2015

A036964 Primes with digits (0,...,8) taken as base 9 and converted to base 10.

Original entry on oeis.org

2, 3, 5, 7, 10, 12, 16, 21, 28, 34, 37, 39, 43, 48, 55, 61, 64, 66, 75, 82, 84, 88, 93, 106, 109, 115, 127, 133, 138, 142, 147, 154, 172, 183, 187, 192, 199, 208, 214, 219, 226, 232, 235, 237, 250, 253, 255, 259, 271, 277, 286, 291, 304, 309, 318, 325, 343, 352

Offset: 1

Patrick De Geest, Jan 04 1999

			a(n)=343 -> is 421{9} -> 421{10} is prime.

Cf. A038617, A036952-A036963.

Offset 1 from Michel Marcus, Oct 10 2019

A235475 Primes whose base-2 representation also is the base-5 representation of a prime.

Original entry on oeis.org

2, 7, 11, 13, 19, 41, 59, 127, 151, 157, 167, 173, 181, 191, 223, 233, 241, 271, 313, 331, 409, 421, 443, 463, 541, 563, 577, 607, 613, 641, 701, 709, 733, 743, 809, 859, 877, 919, 929, 953, 967, 991, 1021, 1033, 1069, 1087, 1193, 1259, 1373, 1423, 1451, 1453, 1471, 1483, 1493, 1549, 1697, 1753, 1759, 1783, 1787, 1831, 1877, 1979, 1993

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.

			7 = 111_2 and 111_5 = 31 are both prime, so 7 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235266, A152079, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,2],5]]&] (* Harvey P. Dale, Jun 15 2019 *)

is(p,b=5,c=2)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ This code is only valid for b>c.

A267769 Numbers whose base-9 representation is a square when read in base 10.

Original entry on oeis.org

0, 1, 4, 15, 23, 33, 58, 73, 81, 100, 121, 185, 213, 265, 298, 324, 361, 400, 474, 509, 555, 643, 685, 751, 861, 914, 1093, 1153, 1215, 1288, 1354, 1481, 1554, 1705, 1783, 1863, 1945, 2029, 2210, 2301, 2488, 2584, 2673, 2773, 2875, 3101, 3210, 3424, 3538, 3682, 3802, 4038, 4154, 4281, 4450

Offset: 1

M. F. Hasler, Jan 20 2016

Trivially includes powers of 81, since 81^k = 100..00_9 = 10^(2k) when read in base 10. Moreover, for any a(n) in the sequence, 81*a(n) is also in the sequence. One could call "primitive" the terms not of this form. These primitive terms include the subsequence 81^k + 2*9^k + 1 = (9^k+1)^2, k > 0, which yields A033934 when written in base 9.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A267763 - A267768 for bases 3 through 8. The base-2 analog is A000302 = powers of 4.

For a "prime" analog see also A235265, A235266, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924.

Select[Range[0, 5000], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 9] &] (* Michael De Vlieger, Jan 24 2016 *)

is(n,b=9,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

A267769_list = [int(s, 9) for s in (str(i**2) for i in range(10**6)) if max(s) < '9'] # Chai Wah Wu, Jan 20 2016

A235473 Primes whose base-3 representation is also the base-4 representation of a prime.

Original entry on oeis.org

2, 43, 61, 67, 97, 103, 127, 139, 151, 157, 199, 211, 229, 277, 283, 331, 337, 349, 373, 379, 433, 439, 463, 499, 523, 571, 601, 607, 727, 751, 787, 823, 853, 883, 919, 991, 1063, 1087, 1117, 1213, 1249, 1327, 1381, 1429, 1483, 1531, 1567, 1597, 1627, 1759, 1783, 1867, 1999

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

This is a subsequence of A045331 and A045375.

			43 = 1121_3 and 1121_4 = 89 are both prime, so 43 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235266, A235474, A152079, A235475 - A235479, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,3],4]]&] (* Harvey P. Dale, Oct 16 2015 *)

is(p,b=4,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A267763 Numbers whose base-3 representation is a square when read in base 10.

Original entry on oeis.org

0, 1, 9, 16, 81, 100, 144, 235, 729, 784, 900, 961, 1296, 1369, 2115, 6561, 6724, 7056, 7225, 8100, 8649, 11664, 11881, 12321, 15985, 19035, 59049, 59536, 60516, 61009, 63504, 64009, 65025, 72900, 73441, 77841, 104976, 105625, 106929, 110889, 143865, 171315, 182428, 531441, 532900, 535824, 537289, 544644, 546121

Offset: 1

M. F. Hasler, Jan 20 2016

Trivially includes powers of 9, since 9^k = 100..00_3 = 10^(2k) when read as a base-10 number. Moreover, for any a(n) in the sequence, 9*a(n) is also in the sequence. One could call "primitive" the terms not of this form; these would be 1, 16 = 121_3, 100 = 10201_3, 235 = 22201_3, 784 = 1002001_3, 961 = 1022121_3, ... These primitive terms include the subsequence 9^k + 2*3^k + 1, k > 0, which yields A033934 when written in base 3.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A267764 - A267769 for bases 4 through 9. The base-2 analog is A000302 = powers of 4.

For a "prime" analog see also A235265, A235266, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924.

[n: n in [0..10^6] | IsSquare(Seqint(Intseq(n, 3)))]; // Vincenzo Librandi, Dec 28 2016

Select[Range[0, 600000], IntegerQ@Sqrt@FromDigits@IntegerDigits[#, 3] &] (* Vincenzo Librandi Dec 28 2016 *)

is(n,b=3,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

A267763_list = [int(d,3) for d in (str(i**2) for i in range(10**6)) if max(d) < '3'] # Chai Wah Wu, Mar 12 2016

cp's OEIS Frontend

A020449 Primes whose greatest digit is 1.

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A065720 Primes whose binary representation is also the decimal representation of a prime.

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A235265 Primes whose base-3 representation also is the base-2 representation of a prime.

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A235482 Primes whose base-5 representation is also the base-9 representation of a prime.

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A036953 Primes having only {0, 1, 2} as digits.

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A036964 Primes with digits (0,...,8) taken as base 9 and converted to base 10.

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A235475 Primes whose base-2 representation also is the base-5 representation of a prime.

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