A089971 -id:A089971 - cp's OEIS frontend

A235629 Primes whose base-9 representation also is the base-5 representation of a prime.

2, 3, 11, 19, 29, 31, 101, 109, 181, 191, 199, 281, 337, 739, 751, 769, 811, 821, 839, 919, 929, 991, 1459, 1489, 1549, 1721, 1741, 1811, 2269, 2281, 2371, 2389, 2441, 2459, 2531, 2539, 2551, 2953, 3089, 3109, 3251, 3271, 6571, 6599, 6661, 6907, 7309, 7321, 7489, 7537, 8039

Offset: 1

M. F. Hasler, Jan 13 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.

			Both 11 = 12_9 and 12_5 = 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. A235482, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235639. See the LINK for further cross-references.

pr95Q[n_]:=Module[{idn9=IntegerDigits[n,9]},Max[idn9]<5&&PrimeQ[ FromDigits[ idn9,5]]]; Select[Prime[Range[1100]],pr95Q] (* Harvey P. Dale, Nov 30 2022 *)

is(p,b=5,c=9)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,9,5)&&print1(vector(#d=digits(p,5),i,9^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,5,9)

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

A281299 Primes p whose binary representation p_2 is the decimal representation of a prime q; and also the sum of the decimal digits of p equals the sum of the digits of p_2.

Original entry on oeis.org

5011, 7001, 11251, 22501, 32303, 32411, 90031, 101107, 104123, 108011, 111323, 121343, 122131, 124001, 125101, 141023, 224011, 233021, 235003, 241141, 321203, 324011, 421303, 432031, 442201, 510331, 511213, 520411, 801011, 1000183, 1000541, 1001191, 1005223, 1006231

Offset: 1

K. D. Bajpai, Jan 19 2017

Intersection of A037308 and A065720.

			a(1) = 5011 is a prime;
5011_2 = 1001110010011_10 is a prime;
5 + 0 + 1 + 1 = 7;
1 + 0 + 0 + 1 + 1 + 1 + 0 + 0 + 1 + 0 + 0 + 1 + 1 = 7; both the digit sums are equal.

Cf. A000040, A033548, A037308, A065720, A089971.

Select[Prime[Range[1000000]], PrimeQ[FromDigits[IntegerDigits[#, 2]]] && Plus @@ IntegerDigits[#] == Plus @@ IntegerDigits[FromDigits[IntegerDigits[#, 2]]] &]

eva(n) = subst(Pol(n), x, 10)
is(n) = ispseudoprime(n) && ispseudoprime(eva(binary(n))) && sumdigits(n)==sumdigits(eva(binary(n))) \\ Felix Fröhlich, Jan 19 2017

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A235629 Primes whose base-9 representation also is the base-5 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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A281299 Primes p whose binary representation p_2 is the decimal representation of a prime q; and also the sum of the decimal digits of p equals the sum of the digits of p_2.

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