A036952 -id:A036952 - cp's OEIS frontend

A235478 Primes whose base-2 representation also is the base-8 representation of a prime.

7, 11, 13, 29, 37, 43, 47, 53, 61, 67, 71, 73, 107, 139, 149, 199, 211, 227, 263, 293, 307, 311, 317, 331, 347, 383, 389, 421, 461, 467, 541, 593, 601, 619, 641, 643, 739, 811, 863, 907, 937, 1061, 1069, 1093, 1117, 1163, 1223, 1283, 1301, 1319, 1321, 1409, 1433, 1439, 1489, 1499, 1523, 1559, 1619, 1697, 1811, 1861, 1879

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.
Appears to be a subsequence of A050150, A062090 and A216285.

			11 = 1011_2 and 1011_8 = 521 are both prime, so 11 is a term.

Robert Price, Table of n, a(n) for n = 1..2265
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235465 ⊂ A077722, A235266, A152079, A235475 - A235479, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[300]],PrimeQ[FromDigits[IntegerDigits[#,2],8]]&] (* Harvey P. Dale, Sep 25 2015 *)

is(p,b=8)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

A235481 Primes whose base-4 representation is also the base-9 representation of a prime.

Original entry on oeis.org

2, 3, 29, 41, 61, 89, 109, 149, 157, 281, 293, 313, 401, 421, 433, 593, 701, 709, 1013, 1049, 1061, 1069, 1097, 1117, 1277, 1289, 1301, 1553, 1601, 1709, 2069, 2137, 2237, 2309, 2377, 2437, 2477, 2689, 2729, 2749, 2797, 2957, 2969, 3001, 3061, 3109, 3169, 3329, 3361, 3389, 3457, 3533, 3701

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.

Appears to be a subsequence of A197636.

			29 = 131_4 and 131_9 = 109 are both prime, so 29 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. A235473 - A235480, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime@Range@600, PrimeQ[FromDigits[IntegerDigits[#, 4], 9]] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=9,c=4)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: Code only valid for b > c.

A267764 Numbers whose base-4 representation is a square when read in base 10.

Original entry on oeis.org

0, 1, 16, 25, 256, 289, 400, 441, 673, 1761, 1849, 4096, 4225, 4624, 4761, 6400, 6561, 7056, 7713, 10768, 13401, 28176, 29584, 65536, 66049, 67600, 68121, 73984, 74529, 76176, 76729, 77985, 102400, 103041, 104976, 112896, 113569, 123408, 150081, 172288, 214416, 450816, 473344, 501433, 519873

Offset: 1

M. F. Hasler, Jan 20 2016

Trivially includes powers of 16, since 16^k = 100..00_4 = 10^(2k) when read as a base-10 number. Moreover, for any a(n) in the sequence, 16*a(n) is also in the sequence. One could call "primitive" the terms not of this form, these would be 1, 25 = 121_4, 289 = 10201_4, 441 = 12321_4, 673 = 22201_4, 1761 = 123201_4, ... These primitive terms include the subsequence 16^k + 2*4^k + 1 = (4^k+1)^2, k > 0, which yields A033934 when written in base 4.

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

Cf. A267763 - A267769 for bases 3 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.

Select[Range[1000], IntegerQ[Sqrt[FromDigits[IntegerDigits[#, 4]]]] &] (* Alonso del Arte, Jan 23 2016 *)

is(n,b=4,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

A267764_list = [int(d,4) for d in (str(i**2) for i in range(10**6)) if max(d) < '4'] # Chai Wah Wu, Feb 23 2016

A267768 Numbers whose base-8 representation is a square when read in base 10.

Original entry on oeis.org

0, 1, 4, 14, 21, 30, 52, 64, 81, 100, 149, 174, 212, 241, 256, 289, 382, 405, 446, 532, 622, 661, 804, 849, 896, 1012, 1045, 1102, 1220, 1281, 1344, 1409, 1476, 1557, 1630, 1780, 1920, 2001, 2197, 2286, 2452, 2545, 2593, 2878, 2965, 3070, 3233, 3328, 3441, 3540, 3630, 3733, 4068, 4096

Offset: 1

M. F. Hasler, Jan 20 2016

Trivially includes powers of 64, since 64^k = 100..00_8 = 10^(2k) when read in base 10. Moreover, for any a(n) in the sequence, 64*a(n) is also in the sequence. One could call "primitive" the terms not of this form. These primitive terms include the subsequence 64^k + 2*8^k + 1 = (8^k+1)^2, k > 0, which yields A033934 when written in base 8.

Motivated by the subsequence A267490 which lists the primes in this sequence.

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

Cf. A267763 - A267769 for bases 3 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..10^4] | IsSquare(Seqint(Intseq(n, 8)))]; // Vincenzo Librandi, Dec 28 2016

Select[Range[0, 2 10^4], IntegerQ@Sqrt@FromDigits@IntegerDigits[#, 8] &] (* Vincenzo Librandi, Dec 28 2016 *)

is(n,b=8,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

A267768_list = [int(s, 8) for s in (str(i**2) for i in range(10**6)) if max(s) < '8'] # Chai Wah Wu, Jan 20 2016

A036961 Primes with digits (0,...,6) taken as base 7 and converted to base 10.

Original entry on oeis.org

2, 3, 5, 8, 10, 17, 22, 29, 31, 38, 43, 50, 52, 59, 71, 85, 94, 106, 115, 122, 127, 134, 143, 155, 157, 169, 185, 197, 211, 218, 220, 227, 239, 241, 248, 260, 262, 274, 290, 295, 304, 316, 323, 325, 332, 337, 353, 358, 365, 367, 379, 386, 388, 395, 409, 428

Offset: 1

Patrick De Geest, Jan 04 1999

			a(n)=323 -> is 641{7} -> 641{10} is prime.

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

Cf. A036952-A036964.

pd7[n_]:=With[{c=IntegerDigits[n]},If[Max[c]<7,FromDigits[c,7],Nothing]]; pd7/@Prime[Range[300]] (* Harvey P. Dale, Mar 14 2025 *)

A036962 Primes without {8, 9} as digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 101, 103, 107, 113, 127, 131, 137, 151, 157, 163, 167, 173, 211, 223, 227, 233, 241, 251, 257, 263, 271, 277, 307, 311, 313, 317, 331, 337, 347, 353, 367, 373, 401, 421, 431, 433, 443, 457, 461

Offset: 1

Patrick De Geest, Jan 04 1999

Harvey P. Dale, Table of n, a(n) for n = 1..10000
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 A038617.

Cf. A036952-A036964.

Select[FromDigits/@Tuples[Range[0,7],3],PrimeQ] (* Harvey P. Dale, Apr 13 2017 *)

A036963 Primes with digits (0,...,7) taken as base 8 and converted to base 10.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 15, 19, 25, 31, 33, 35, 39, 43, 49, 55, 57, 59, 65, 67, 71, 75, 87, 89, 95, 105, 111, 115, 119, 123, 137, 147, 151, 155, 161, 169, 175, 179, 185, 191, 199, 201, 203, 207, 217, 223, 231, 235, 247, 251, 257, 273, 281, 283, 291, 303, 305, 307, 311

Offset: 1

Patrick De Geest, Jan 04 1999

			a(n)=191 -> is 277{8} -> 277{10} is prime.

Cf. A036952-A036964.

Offset 1 from Michel Marcus, Oct 10 2019

A156059 Composite numbers whose binary representation reads as decimal prime.

Original entry on oeis.org

185, 247, 253, 295, 329, 355, 405, 425, 453, 471, 533, 539, 565, 583, 595, 671, 675, 689, 703, 755, 781, 785, 815, 841, 855, 925, 989, 1037, 1075, 1099, 1113, 1121, 1159, 1189, 1207, 1219, 1269, 1287, 1305, 1329, 1341, 1403, 1413, 1441, 1473, 1521, 1541

Offset: 1

Gerald Hillier, Feb 03 2009

			185 is composite & 10111001 is prime.
313 is not in the sequence as it is prime in base 10.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A065720, A007088, A036952.

Select[Range[2900], ! PrimeQ[#]&&PrimeQ[FromDigits[IntegerDigits[#, 2]]]&] (* Vincenzo Librandi, Apr 18 2013 *)

More terms from R. J. Mathar, Feb 10 2009

Edited by N. J. A. Sloane, Mar 17 2010

A174416 Numbers whose binary expansion is a decimal nonprime.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Jaroslav Krizek, Mar 19 2010

Complement of A036952.

			For n=6, a(6) = 8; binary expansion of 8 = 1000 (nonprime).

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

Cf. A036952, A174417 (binary expansion).

remove(t -> isprime(convert(t,binary)), [$1..100]); # Robert Israel, Jun 13 2017

Select[Range[80],!PrimeQ[FromDigits[IntegerDigits[#,2]]]&] (* Harvey P. Dale, May 10 2012 *)

Corrected and extended by Harvey P. Dale, May 10 2012

A231478 Primes whose base-3 representation is also the base-8 representation of a prime.

Original entry on oeis.org

2, 7, 13, 31, 37, 43, 67, 73, 97, 193, 283, 307, 379, 457, 487, 499, 577, 619, 643, 727, 733, 757, 829, 1297, 1321, 1429, 1447, 1609, 1669, 1693, 2011, 2083, 2137, 2251, 2269, 2347, 2539, 2803, 2857, 2953, 2971

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.

			13 = 111_3 and 111_8 = 73 are both prime, so 13 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. A235471, A235265, A235266, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924. See the LINK for further cross-references.

Select[Prime@Range@500, PrimeQ@FromDigits[IntegerDigits[#, 3], 8] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=8,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

cp's OEIS Frontend

A235478 Primes whose base-2 representation also is the base-8 representation of a prime.

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A235481 Primes whose base-4 representation is also the base-9 representation of a prime.

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A267764 Numbers whose base-4 representation is a square when read in base 10.

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A267768 Numbers whose base-8 representation is a square when read in base 10.

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A036961 Primes with digits (0,...,6) taken as base 7 and converted to base 10.

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A036962 Primes without {8, 9} as digits.

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A036963 Primes with digits (0,...,7) taken as base 8 and converted to base 10.

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A156059 Composite numbers whose binary representation reads as decimal prime.

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