A152079 -id:A152079 - cp's OEIS frontend

A235615 Primes whose base-5 representation also is the base-4 representation of a prime.

2, 3, 13, 41, 43, 61, 181, 191, 263, 281, 283, 331, 383, 431, 443, 463, 641, 643, 661, 881, 911, 1063, 1091, 1291, 1303, 1531, 1693, 2083, 2143, 2203, 2293, 2341, 3163, 3181, 3191, 3253, 3343, 3593, 3761, 3931, 4001, 4093, 4391, 4691, 4793, 5011, 5393, 5413, 5441, 6301

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 13 = 23_5 and 23_4 = 11 are prime.

Robert Price, Table of n, a(n) for n = 1..107268
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235474, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235639. See the LINK for further cross-references.

is(p,b=4,c=5)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,5,4)&&print1(vector(#d=digits(p,4),i,5^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,4,5)

A235639 Primes whose base-9 representation is also the base-6 representation of a prime.

Original entry on oeis.org

2, 3, 5, 19, 23, 41, 113, 127, 131, 163, 199, 271, 419, 433, 739, 743, 761, 919, 991, 1009, 1013, 1063, 1153, 1171, 1459, 1481, 1499, 1553, 1567, 1571, 1733, 1747, 1783, 1873, 1913, 2237, 2377, 2381, 2539, 2557, 2593, 2633, 2939, 3011, 3079, 3083, 3187, 3259, 3331, 3659

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.

			19 = 21_9 and 21_6 = 13 are both prime, so 19 is a term.
509 = 625_9 and 625_6 = 17 are both prime, but 625 is not a valid base-6 integer, so 509 is not a term.

Robert Israel, 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. A231481, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235638. See the LINK for further cross-references.

R:= 2: x:= 2: count:= 1:
while count < 100 do
  x:= nextprime(x);
  L:= convert(x,base,6);
  y:= add(9^(i-1)*L[i],i=1..nops(L));
  if isprime(y) then count:= count+1; R:= R, y fi
od:
R; # Robert Israel, May 18 2020

is(p,b=6,c=9)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,9,6)&&print1(vector(#d=digits(p,6),i,9^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,6,9)

A235475 Primes whose base-2 representation also is the base-5 representation of a prime.

Original entry on oeis.org

2, 7, 11, 13, 19, 41, 59, 127, 151, 157, 167, 173, 181, 191, 223, 233, 241, 271, 313, 331, 409, 421, 443, 463, 541, 563, 577, 607, 613, 641, 701, 709, 733, 743, 809, 859, 877, 919, 929, 953, 967, 991, 1021, 1033, 1069, 1087, 1193, 1259, 1373, 1423, 1451, 1453, 1471, 1483, 1493, 1549, 1697, 1753, 1759, 1783, 1787, 1831, 1877, 1979, 1993

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.

			7 = 111_2 and 111_5 = 31 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. A235266, A152079, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,2],5]]&] (* Harvey P. Dale, Jun 15 2019 *)

is(p,b=5,c=2)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ This code is only valid for b>c.

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

Original entry on oeis.org

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

A235473 Primes whose base-3 representation is also the base-4 representation of a prime.

Original entry on oeis.org

2, 43, 61, 67, 97, 103, 127, 139, 151, 157, 199, 211, 229, 277, 283, 331, 337, 349, 373, 379, 433, 439, 463, 499, 523, 571, 601, 607, 727, 751, 787, 823, 853, 883, 919, 991, 1063, 1087, 1117, 1213, 1249, 1327, 1381, 1429, 1483, 1531, 1567, 1597, 1627, 1759, 1783, 1867, 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.

This is a subsequence of A045331 and A045375.

			43 = 1121_3 and 1121_4 = 89 are both prime, so 43 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. A235266, A235474, A152079, A235475 - A235479, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,3],4]]&] (* Harvey P. Dale, Oct 16 2015 *)

is(p,b=4,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A267763 Numbers whose base-3 representation is a square when read in base 10.

Original entry on oeis.org

0, 1, 9, 16, 81, 100, 144, 235, 729, 784, 900, 961, 1296, 1369, 2115, 6561, 6724, 7056, 7225, 8100, 8649, 11664, 11881, 12321, 15985, 19035, 59049, 59536, 60516, 61009, 63504, 64009, 65025, 72900, 73441, 77841, 104976, 105625, 106929, 110889, 143865, 171315, 182428, 531441, 532900, 535824, 537289, 544644, 546121

Offset: 1

M. F. Hasler, Jan 20 2016

Trivially includes powers of 9, since 9^k = 100..00_3 = 10^(2k) when read as a base-10 number. Moreover, for any a(n) in the sequence, 9*a(n) is also in the sequence. One could call "primitive" the terms not of this form; these would be 1, 16 = 121_3, 100 = 10201_3, 235 = 22201_3, 784 = 1002001_3, 961 = 1022121_3, ... These primitive terms include the subsequence 9^k + 2*3^k + 1, k > 0, which yields A033934 when written in base 3.

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

Cf. A267764 - A267769 for bases 4 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..10^6] | IsSquare(Seqint(Intseq(n, 3)))]; // Vincenzo Librandi, Dec 28 2016

Select[Range[0, 600000], IntegerQ@Sqrt@FromDigits@IntegerDigits[#, 3] &] (* Vincenzo Librandi Dec 28 2016 *)

is(n,b=3,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

A267763_list = [int(d,3) for d in (str(i**2) for i in range(10**6)) if max(d) < '3'] # Chai Wah Wu, Mar 12 2016

A235479 Primes whose base-2 representation also is the base-9 representation of a prime.

Original entry on oeis.org

11, 13, 19, 41, 79, 109, 137, 151, 167, 191, 193, 199, 227, 239, 271, 307, 313, 421, 431, 433, 457, 487, 491, 521, 563, 613, 617, 659, 677, 709, 727, 757, 929, 947, 1009, 1033, 1051, 1249, 1483, 1693, 1697, 1709, 1721, 1831, 1951, 1979, 1987, 1993

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.
A subsequence of A027697, A050150, A062090 and A176620.

			11 = 1011_2 and 1011_9 = 6571 are both prime, so 11 is a term.

Robert Price, Table of n, a(n) for n = 1..1958
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235466 ⊂ A077723, A235266, A152079, A235475 - A235478, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

is(p,b=9)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

A267490 Primes whose base-8 representation is a perfect square in base 10.

Original entry on oeis.org

149, 241, 661, 1409, 2593, 3733, 6257, 7793, 15313, 23189, 25601, 26113, 30497, 34337, 44053, 49057, 78577, 92821, 95009, 108529, 115861, 132757, 162257, 178417, 183377, 223381, 235541, 242197, 266261, 327317, 345749, 426389, 525461, 693397, 719893, 729713, 805397, 814081, 903841

Offset: 1

Christopher Cormier, Jan 16 2016

Subsequence of primes in A267768. - M. F. Hasler, Jan 20 2016

			a(1) = 149 because 149 is 225 in base 8, and 225 is 15^2 in base 10.

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

For primes which are primes in other bases, see A235265, A235266, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924.

[n:n in PrimesUpTo(1000000)| IsSquare(Seqint(Intseq(n,8)))]; // Marius A. Burtea, Jun 30 2019

Select[Prime@ Range[10^5], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 8] &] (* Michael De Vlieger, Jan 16 2016 *)

listp(nn) = {forprime(p=1, nn, d = digits(p, 8); pd = Pol(d); if (issquare(subst(pd, x, 10)), print1(p, ", ")););} \\ Michel Marcus, Jan 16 2016

is(n,b=8,c=10)={issquare(subst(Pol(digits(n,b)),x,c))&&isprime(n)} \\ M. F. Hasler, Jan 20 2016

from sympy import isprime
A267490_list = [int(s,8) for s in (str(i**2) for i in range(10**6)) if max(s) < '8' and isprime(int(s,8))] # Chai Wah Wu, Jan 20 2016

A235474 Primes whose base-4 representation is also the base-5 representation of a prime.

Original entry on oeis.org

2, 3, 11, 29, 31, 41, 101, 109, 139, 149, 151, 181, 199, 229, 239, 251, 269, 271, 281, 389, 409, 491, 509, 541, 547, 661, 751, 887, 911, 947, 991, 1021, 1051, 1061, 1069, 1091, 1151, 1279, 1289, 1381, 1409, 1471, 1549, 1709, 1759, 1801, 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.

For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.

			11 = 23_4 and 23_5 = 13 are both prime, so 11 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. A235266, A235473, A152079, A235475 - A235479, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,4],5]]&] (* Harvey P. Dale, Dec 31 2017 *)

is(p,b=5,c=4)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A235477 Primes whose base-2 representation also is the base-7 representation of a prime.

Original entry on oeis.org

2, 31, 47, 59, 103, 107, 173, 179, 181, 199, 211, 227, 229, 233, 367, 409, 443, 463, 487, 701, 743, 757, 823, 827, 877, 911, 919, 967, 1009, 1123, 1163, 1291, 1321, 1367, 1373, 1447, 1493, 1571, 1583, 1597, 1609, 1627, 1657, 1669, 1721, 1831, 1933, 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.
A subsequence of A027697, A015919, A197636 (conjectural).

			31 = 11111_2 and 11111_7 = 2801 are both prime, so 31 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. A235464 ⊂ A077721, 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],7]]&] (* Harvey P. Dale, May 08 2021 *)

is(p,b=7)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

cp's OEIS Frontend

A235615 Primes whose base-5 representation also is the base-4 representation of a prime.

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A235639 Primes whose base-9 representation is also the base-6 representation of a prime.

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Maple

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A235475 Primes whose base-2 representation also is the base-5 representation of a prime.

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Mathematica

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

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A235473 Primes whose base-3 representation is also the base-4 representation of a prime.

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A267763 Numbers whose base-3 representation is a square when read in base 10.

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Magma

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A235479 Primes whose base-2 representation also is the base-9 representation of a prime.

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A267490 Primes whose base-8 representation is a perfect square in base 10.