A267763 -id:A267763 - cp's OEIS frontend

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

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

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

A267765 Numbers whose base-5 representation is a square when read in base 10.

Original entry on oeis.org

0, 1, 4, 25, 36, 49, 89, 100, 121, 139, 249, 329, 351, 625, 676, 729, 900, 961, 1225, 1551, 1654, 2146, 2225, 2289, 2500, 2601, 3025, 3289, 3475, 3521, 3814, 4324, 4529, 4801, 5086, 5149, 6225, 6726, 6829, 7374, 8225, 8464, 8775, 9454, 9601, 13926, 15625, 15876, 16129, 16900, 17161

Offset: 1

M. F. Hasler, Jan 20 2016

Trivially includes powers of 25, since 25^k = 100..00_5 = 10^(2k) when read in base 10. Moreover, for any a(n) in the sequence, 25*a(n) is also in the sequence. One could call "primitive" the terms not of this form, these would be 1, 4, 36 = 121_5, 49 = 144_5, 89 = 324_5, ... These primitive terms include the subsequence 25^k + 2*5^k + 1 = (5^k+1)^2, k > 0, which yields A033934 when written in base 5.

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[0, 17200], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 5] &] (* Michael De Vlieger, Jan 24 2016 *)

is(n,b=5,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

A267765_list = [int(d,5) for d in (str(i**2) for i in range(10**6)) if max(d) < '5'] # Chai Wah Wu, Mar 12 2016

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

Original entry on oeis.org

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

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

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

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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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A267765 Numbers whose base-5 representation is a square when read in base 10.

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

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

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