A065727 -id:A065727 - cp's OEIS frontend

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

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

A281252 Numbers whose septenary, octal and nonary representations are prime when read in decimal.

Original entry on oeis.org

2, 3, 5, 43, 115, 619, 1249, 1681, 1711, 2563, 2635, 5155, 10321, 10531, 11539, 13219, 14479, 17713, 17755, 18217, 18889, 20203, 20905, 26335, 27163, 29305, 35353, 39859, 40867, 40897, 40993, 44425, 44803, 51145, 52993, 55735, 57751, 58075, 68335, 68839, 69553

Offset: 1

K. D. Bajpai, Jan 22 2017

After a(1) all the terms are odd.

After a(2) all terms are relatively prime to 42. - Charles R Greathouse IV, Jan 22 2017

			a(6) = 619 is in the sequence because 619_10 = 1543_7 = 1153_8 = 757_9; and 1543, 1153 and 757 are prime when read in decimal.
a(7) = 1249 is in the sequence because 1249_10 = 3433_7 = 2341_8 = 1637_9; and 3433, 2341 and 1637 are prime when read in decimal.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A065726, A065727, A138299, A175279, A236365.

Select[Range[500000], PrimeQ[FromDigits[IntegerDigits[#, 7]]] && PrimeQ[FromDigits[IntegerDigits[#, 8]]] && PrimeQ[FromDigits[IntegerDigits[#, 9]]] &]

is(n)=for(b=7,9, if(!isprime(fromdigits(digits(n,b))), return(0))); 1 \\ Charles R Greathouse IV, Jan 22 2017

A065425 Smallest odd prime greater than n such that the decimal expansion of its base n conversion is also prime.

Original entry on oeis.org

3, 67, 5, 13, 7, 17, 11, 37, 11, 31, 13, 29, 17, 41, 17, 37, 19, 41, 23, 53, 23, 113, 31, 53, 29, 109, 29, 61, 31, 71, 41, 89, 37, 73, 37, 83, 41, 89, 41, 109, 43, 89, 47, 101, 47, 97, 61, 101, 53, 113, 53, 109, 61, 113, 59, 137, 59, 127, 61, 131, 71, 137, 67, 139, 67, 137

Offset: 2

Robert G. Wilson v, Nov 15 2001

For bases above ten, use multidigit numbers to represent the new number. See the example.

			a(15) = 41. 17d = 12, 19d = 14, 23d = 18, 29d = 114, 31d = 21 and 37d = 27, all of which are composite. But 41d = 211 which is a prime.

Cf. A065720 - A065727.

Do[ k = PrimePi[n] + 1; While[ !PrimeQ[ FromDigits[ IntegerDigits[ Prime[k], n]]], k++ ]; Print[ Prime[k]], {n, 2, 50} ]

More terms from David Wasserman, Aug 30 2002

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

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A281252 Numbers whose septenary, octal and nonary representations are prime when read in decimal.

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A065425 Smallest odd prime greater than n such that the decimal expansion of its base n conversion is also prime.

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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.)

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