A267490 -id:A267490 - cp's OEIS frontend

A267821 Primes whose base-9 representation is a square in base 10.

23, 73, 509, 643, 751, 1093, 1153, 1481, 1783, 2029, 6067, 8011, 8933, 9613, 16061, 24763, 30803, 44203, 64633, 74521, 82723, 88003, 94033, 103171, 106619, 118669, 126541, 133321, 154571, 158293, 163561, 165883, 179821, 197339, 202393, 207643, 225523, 234733, 264083, 268003, 284593, 288661, 324931

Offset: 1

Christopher Cormier, Jan 20 2016

Primes in the sequence A267769.

			73 in base 9 is 81, and 81 = 9^2 in base 10.

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

In other bases: A267817 - A267821, A241246, A267490.

Select[Prime@ Range@ 28000, IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 9] &] (* Michael De Vlieger, Jan 24 2016 *)

from sympy import isprime
A267821_list = [int(d,9) for d in (str(i**2) for i in range(1,10**6)) if max(d) < '9' and isprime(int(d,9))] # Chai Wah Wu, Feb 22 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

A267819 Primes whose base-5 representation is a square when read in base 10.

Original entry on oeis.org

89, 139, 4801, 6829, 9601, 48661, 305551, 539849, 867001, 3503449, 3527071, 6063839, 13912039, 15933439, 18100351, 18319321, 20366399, 22849121, 26239321, 46724801, 63565049, 78057911, 95802799, 100294871, 102369439, 107474911, 139769449, 141118079, 144829879, 159468079, 161483801

Offset: 1

Christopher Cormier, Jan 20 2016

Primes in the sequence A267765.

			89 in base 5 is 324, and 324 = 18^2 in base 10.

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

In other bases: A267817 - A267821, A241246, A267490.

Select[Prime@ Range[10^6], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 5] &] (* Michael De Vlieger, Jan 24 2016 *)

from sympy import isprime
A267819_list = [int(d,5) for d in (str(i**2) for i in range(1,10**6)) if max(d) < '5' and isprime(int(d,5))] # Chai Wah Wu, Feb 22 2016

A267820 Primes whose base-6 representation is a square when read in base 10.

Original entry on oeis.org

17, 89, 409, 449, 577, 953, 7649, 12401, 14593, 23689, 28393, 29033, 30689, 36809, 40153, 46457, 47969, 54577, 56393, 59273, 65033, 72649, 78713, 92033, 98953, 106033, 141353, 146249, 150209, 169657, 176489, 199889, 205433, 221849, 235273, 238481, 245033, 258569, 381841, 394633, 441193, 529273

Offset: 1

Christopher Cormier, Jan 20 2016

Primes in the sequence A267766.

			17 in base 6 is 25, and 25 = 5^2 in base 10.

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

In other bases: A267817 - A267821, A241246, A267490.

Select[Prime@ Range@ 45000, IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 6] &] (* Michael De Vlieger, Jan 24 2016 *)

from sympy import isprime
A267820_list = [int(d,6) for d in (str(i**2) for i in range(1,10**6)) if max(d) < '6' and isprime(int(d,6))] # Chai Wah Wu, Feb 22 2016

A267818 Primes whose base-4 representation is a square when read in base 10.

Original entry on oeis.org

673, 1165921, 6228961, 53378329, 128818873, 805379161, 926711449, 1542173569, 2459671993, 8365652929, 14232980953, 26484614209, 30755887321, 31937961241, 37968797689, 78180076129, 80974979329, 97835803129, 168394504129, 314911422913, 410879716321, 529074406681

Offset: 1

Christopher Cormier, Jan 20 2016

Primes in the sequence A267764.

			673 written in base 4 is 22201, and 22201 in base 10 is 149^2.

Chai Wah Wu, Table of n, a(n) for n = 1..100

In other bases: A267817, A267818, A267819, A267820, A267821, A241246, A267490.

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

is(n, b=4, c=10)=issquare(subst(Pol(digits(n, b)), x, c))
lista(nn) = forprime(p=2, nn, if (is(p), print1(p, ", "))) \\ Michel Marcus, Jan 24 2016

from sympy import isprime
A267818_list = [int(d,4) for d in (str(i**2) for i in range(1,10**6)) if max(d) < '4' and isprime(int(d,4))] # Chai Wah Wu, Feb 22 2016

a(8)-a(22) from Chai Wah Wu, Feb 22 2016

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A267821 Primes whose base-9 representation is a square 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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A267819 Primes whose base-5 representation is a square when read in base 10.

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A267820 Primes whose base-6 representation is a square when read in base 10.

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A267818 Primes whose base-4 representation is a square when read in base 10.

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