A065721 -id:A065721 - cp's OEIS frontend

A267766 Numbers whose base-6 representation is a square when read in base 10.

0, 1, 4, 17, 36, 49, 64, 89, 124, 144, 169, 232, 305, 388, 409, 449, 544, 577, 612, 665, 953, 1105, 1296, 1369, 1444, 1529, 1764, 1849, 1936, 2033, 2304, 2825, 3097, 3204, 3280, 3473, 4345, 4464, 4588, 4841, 5104, 5184, 5329, 5633, 6084, 6241, 7081, 7649, 8044, 8352, 8449, 9160, 9593

Offset: 1

M. F. Hasler, Jan 20 2016

Trivially includes powers of 36, since 36^k = 100..00_6 = 10^(2k) when read in base 10. Moreover, for any a(n) in the sequence, 36*a(n) is also in the sequence. One could call "primitive" the terms not of this form. These primitive terms include the subsequence 36^k + 2*6^k + 1 = (6^k+1)^2, k > 0, which yields A033934 when written in base 6.

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,6)))]; // Bruno Berselli, Jan 20 2016

Select[Range[0, 10^4], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 6] &] (* Michael De Vlieger, Jan 24 2016 *)

is(n,b=6,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

A267766_list = [int(d,6) for d in (str(i**2) for i in range(10**6)) if max(d) < '6'] # Chai Wah Wu, Mar 12 2016

A281227 Primes whose binary reflected Gray code representation is also the decimal representation of a prime.

Original entry on oeis.org

2, 53, 233, 281, 397, 521, 613, 673, 733, 773, 797, 829, 1049, 1129, 1433, 1553, 1697, 1933, 2129, 2237, 2273, 2281, 2437, 2521, 2557, 2617, 2729, 2969, 3121, 3181, 3413, 3457, 3517, 3637, 3709, 3761, 3881, 4337, 4357, 4729, 4733, 4877, 4889, 5101, 5657, 5813, 5857, 6113, 6133

Offset: 1

Indranil Ghosh, Jan 18 2017

			521 is in the sequence because 521_10 = 1100001101_2 and both 521 and 1100001101 are prime numbers in base 10.

Indranil Ghosh, Table of n, a(n) for n = 1..481

Cf. A014550, A065720, A065721, A065722, A065723, A065724.

from sympy import isprime
def gray(n):
    return bin(n^(n//2))[2:]
i=1
j=1
while j<=481:
    if isprime(i)==True and isprime(int(gray(i)))==True:
        print(str(j)+" "+str(i))
        j+=1
    i+=1

A231476 Primes whose base-3 representation is also the base-6 representation of a prime.

Original entry on oeis.org

2, 7, 13, 19, 31, 151, 163, 211, 223, 229, 241, 271, 349, 367, 439, 601, 607, 613, 631, 643, 673, 727, 733, 859, 907, 937, 997, 1021, 1033, 1039, 1051, 1093, 1117, 1123, 1129, 1153, 1321, 1327, 1399, 1423, 1429, 1609, 1627, 1657, 1669, 1741, 1747, 1759, 1777, 1789, 1831, 1867, 1933, 1951, 1993, 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.

Subsequence of A045375 ⊂ A045331.

			7 = 21_3 and 21_6 = 13 are both prime.

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. A235469, A235265, A235266, A235473, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[ IntegerDigits[#, 3], 6]] &] (* Harvey P. Dale, Sep 29 2016 *)

is(p,b=6,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b>c.

A235464 Primes whose base-7 representation also is the base-2 representation of a prime.

Original entry on oeis.org

7, 2801, 17207, 19559, 134513, 134807, 840743, 842759, 842801, 941249, 943601, 958007, 958049, 958343, 5899657, 6591089, 6607903, 6706393, 6722857, 41196751, 41311663, 41314057, 46137673, 46137967, 46253257, 46942351, 46944409, 47059657

Offset: 1

M. F. Hasler, Jan 11 2014

This sequence is part of the two-dimensional array of sequences 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.
When the smaller base is b=2 such that only digits 0 and 1 are allowed, these are primes that are the sum of distinct powers of the larger base, here c=7, thus a subsequence of A077721.

			7 = 10_7 and 10_2 = 2 are both prime, so 7 is a term.
2801 = 11111_7 and 11111_2 = 31 are both prime, so 2801 is a term.

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235477, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

is(p,b=2,c=7)=vecmax(d=digits(p,c))

forprime(p=1,1e3,is(p,7,2)&&print1(vector(#d=digits(p,2),i,7^(#d-i))*d~,",")) \\ To produce the terms, this is much more efficient than to select them using straightforwardly is(.)=is(.,2,7)

A235468 Primes whose base-5 representation also is the base-3 representation of a prime.

Original entry on oeis.org

2, 5, 7, 11, 31, 37, 127, 131, 151, 157, 257, 281, 311, 661, 677, 751, 757, 877, 881, 907, 911, 1277, 1301, 1381, 1511, 1531, 3137, 3187, 3251, 3307, 3407, 3761, 3877, 3911, 3931, 4001, 4007, 4027, 4051, 4057, 4561, 4637, 6257, 6287, 7057, 7151, 7177, 7187, 7507

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of the two-dimensional array of sequences 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.

			a(1) = 5 = 10_5 and 10_3 = 3 are both prime.
a(2) = 7 = 12_5 and 12_3 = 5 are both prime.
a(3) = 11 = 21_5 and 21_3 = 7 are both prime.

Robert Price, Table of n, a(n) for n = 1..11369
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

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

is(p,b=3,c=5)=vecmax(d=digits(p,c))

forprime(p=1,1e3,is(p,5,3)&&print1(vector(#d=digits(p,3),i,5^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,3,5)

A235472 Primes whose base-9 representation also is the base-3 representation of a prime.

Original entry on oeis.org

2, 11, 19, 83, 101, 163, 173, 739, 811, 821, 829, 911, 1549, 1559, 1621, 6563, 6581, 6661, 6733, 8111, 8191, 13933, 14753, 59069, 59141, 59779, 59797, 59951, 60589, 60607, 65629, 65701, 66359, 67079, 67231, 72271, 72353, 72901, 118189, 119557, 119657, 124669, 124823, 125399

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of the two-dimensional array of sequences 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.
Since all digits of the base 9 expansion are less than 3, this is a subsequence of A037314.

			Both 17 = 21_9 and 21_3 = 7 are prime.

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

is(p,b=3,c=9)=vecmax(d=digits(p,c))

forprime(p=1,2e3,is(p,9,3)&&print1(vector(#d=digits(p,3),i,9^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,3,9)

A235476 Primes whose base-2 representation also is the base-6 representation of a prime.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 29, 41, 53, 67, 101, 127, 193, 263, 281, 337, 353, 431, 461, 479, 487, 499, 523, 593, 599, 631, 743, 757, 773, 821, 823, 829, 857, 883, 887, 941, 1013, 1021, 1093, 1117, 1259, 1279, 1303, 1367, 1373, 1429, 1439, 1459, 1471, 1483, 1493, 1511, 1583, 1619, 1699, 1759, 1831, 1847, 1879, 1931, 1951, 1987

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.

			5 = 101_2 and 101_6 = 37 are both prime, so 5 is a term.
7 = 111_2 and 111_6 = 43 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. A235463 ⊂ A077720, A235475, A152079, A235266, 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],6]]&] (* Harvey P. Dale, Jan 03 2022 *)

is(p,b=6)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

A267767 Numbers whose base-7 representation is a square when read in base 10.

Original entry on oeis.org

0, 1, 4, 13, 19, 27, 46, 49, 64, 81, 117, 139, 165, 190, 196, 225, 313, 361, 433, 460, 571, 603, 637, 705, 748, 837, 883, 931, 981, 1048, 1105, 1222, 1323, 1489, 1560, 1684, 1744, 2028, 2185, 2254, 2346, 2401, 2500, 2601, 2763, 2869, 3084, 3136, 3249, 3364, 3547, 3667, 3865, 3969, 4096

Offset: 1

M. F. Hasler, Jan 20 2016

Trivially includes powers of 49, since 49^k = 100..00_7 = 10^(2k) when read in base 10. Moreover, for any a(n) in the sequence, 49*a(n) is also in the sequence. One could call "primitive" the terms not of this form. These primitive terms include the subsequence 49^k + 2*7^k + 1 = (7^k+1)^2, k > 0, which yields A033934 when written in base 7.

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, 7)))]; // Vincenzo Librandi, Dec 28 2016

Select[Range[0, 2 10^4], IntegerQ@Sqrt@FromDigits@IntegerDigits[#, 7] &] (* Vincenzo Librandi, Dec 28 2016 *)

is(n,b=7,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

A267767_list = [int(s, 7) for s in (str(i**2) for i in range(10**6)) if max(s) < '7'] # Chai Wah Wu, Jan 20 2016

A278931 Semiprimes whose ternary representations are also semiprime when read as a decimal number.

Original entry on oeis.org

25, 49, 65, 82, 106, 115, 118, 121, 142, 143, 155, 187, 209, 235, 254, 259, 262, 265, 274, 289, 299, 314, 319, 326, 334, 335, 341, 355, 361, 382, 398, 415, 445, 451, 454, 458, 469, 493, 511, 515, 538, 551, 562, 566, 583, 586, 589, 614, 622, 634, 649, 667, 679

Offset: 1

K. D. Bajpai, Dec 04 2016

			65 is in the sequence because 5*13 = 65 (semiprime) and its ternary representation, 2102 = 2*1051, when read as a decimal number, is also semiprime.
115 is in the sequence because 5*23 = 115 (semiprime) and its ternary representation, 11021 = 103*107, when read as a decimal number, is also semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..9646
Wikipedia, Ternary numeral system

Subsequence of A001358.

Cf. A001358, A005935, A007089, A065721, A089981, A236537.

Select[Range[5000], PrimeOmega[#] == 2 && PrimeOmega[FromDigits[ IntegerDigits[ #, 3]]] == 2 &]

A316479 a(n) is the smallest prime whose base-b expansion, read as a base-10 number, is a prime for every b in 2, 3, ..., n. (For n > 10, each base-b expansion for 10 < b <= n must contain no digit larger than 9.)

Original entry on oeis.org

3, 157, 157, 9241, 9241, 48404791, 18172964503, 50006393431, 50006393431, 181395559296673

Offset: 2

Jon E. Schoenfield, Jul 16 2018

a(2)=3, the smallest term in A065720, primes whose binary representation is also the decimal representation of a prime;
a(3)=157, the smallest integer in both A065720 and A065721, primes p whose base-3 expansion is also the decimal expansion of a prime;
similarly, a(4)=157 is the smallest integer in A065720, A065721, and A065722.
Is this sequence infinite?
a(12) > 10^16. - Giovanni Resta, Aug 01 2018

			a(2)=3 because 3 is prime, 3_10 = 11_2, and 11 is prime, and 3 is the smallest such number.
a(3)=157 because 157 is prime, 157_10 = 10011101_2, 157_10 = 12211_3, and 10011101 and 12211 are prime, and 157 is the smallest such number. a(4)=157 as well, since 157_10 = 2131_4 and 2131 is also prime.

Cf. A065720, A065721, A065722, A065723, A065724, A065725, A065726, A065727.

Cf. A084482, A236537.

a(8)-a(10) from Giovanni Resta, Jul 17 2018

a(11) from Giovanni Resta, Jul 24 2018

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A267766 Numbers whose base-6 representation is a square when read in base 10.

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A281227 Primes whose binary reflected Gray code representation is also the decimal representation of a prime.

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A231476 Primes whose base-3 representation is also the base-6 representation of a prime.

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A235464 Primes whose base-7 representation also is the base-2 representation of a prime.

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A235468 Primes whose base-5 representation also is the base-3 representation of a prime.

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A235472 Primes whose base-9 representation also is the base-3 representation of a prime.

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A235476 Primes whose base-2 representation also is the base-6 representation of a prime.

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A267767 Numbers whose base-7 representation is a square when read in base 10.

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