A091924 -id:A091924 - cp's OEIS frontend

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

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

A065726 Primes p whose base-8 expansion is also the decimal expansion of a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 31, 43, 59, 67, 71, 89, 137, 151, 179, 191, 199, 223, 251, 257, 281, 283, 307, 311, 337, 353, 359, 367, 383, 409, 419, 433, 443, 449, 523, 563, 617, 619, 641, 659, 727, 787, 809, 811, 857, 887, 907, 919, 947, 977, 1033, 1039, 1097, 1123

Offset: 1

Patrick De Geest, Nov 15 2001

In general rebase notation (Marc LeBrun): p8 = (8) [p] (10).

			E.g., 787_10 = 1423_8 is prime, and so is 1423_10.

Harry J. Smith, 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

Primes in A036963.
Cf. A065720 up to A065727, A065361.
Cf. A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[ Range[2500], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 8]]] & ]

is(p, b=10, c=8)=isprime(vector(#d=digits(p, c), i, b^(#d-i))*d~)&&isprime(p) \\ This code can be used for other bases b, c when b>c. See A235265 for code also valid for bM. F. Hasler, Jan 12 2014

Definition clarified by M. F. Hasler, Jan 12 2014

A090862 Smallest b > 10 such that the decimal representation of the n-th prime is not a prime in base b representation.

Original entry on oeis.org

11, 11, 11, 11, 11, 12, 11, 11, 11, 12, 11, 11, 11, 14, 13, 11, 11, 11, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 11, 11, 12, 11, 11, 11, 11, 11, 11, 12, 12, 11, 11, 12, 11, 11, 11, 11, 11, 11, 12, 11, 11, 11, 12, 11, 11

Offset: 5

Reinhard Zumkeller, Feb 12 2004

			For n = 10: A000040(10) = 29 in base 11 2*11^1 + 9*11^0 = 31 is prime, 29 in base 12 is 2*12^1 + 9*12^0 = 33 = 3*11 is not prime, therefore a(10) = 12.

Amiram Eldar, Table of n, a(n) for n = 5..10000

Cf. A091922, A091923, A091924.

a[n_] := Module[{d = IntegerDigits[Prime[n]], b = 11}, While[PrimeQ[FromDigits[d, b]], b++]; b]; Array[a, 100, 5] (* Amiram Eldar, Mar 28 2025 *)

a(A091922(n)) = n and a(m) <> n for m < A091922(n).

A235110 Primes whose base-10 representation also represents a prime in base 13.

Original entry on oeis.org

2, 3, 5, 7, 23, 41, 47, 61, 83, 89, 157, 173, 179, 197, 223, 229, 263, 281, 311, 313, 331, 373, 379, 397, 401, 463, 467, 487, 571, 599, 607, 643, 661, 739, 751, 773, 797, 809, 823, 863, 883, 919, 937, 971, 977

Offset: 1

M. F. Hasler, Jan 03 2014

See A090712 for a similar sequence whose definition works "in the opposite direction".

			The decimal representation of prime 23, considered as a number written in base 13, stands for 2*13+3 = 29, which is also prime, therefore 23 is in the sequence.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A235110, A235144 and other sequences in the range A090707 - A091924.

Select[Prime[Range[5000]],PrimeQ[FromDigits[IntegerDigits[#],13]]&] (* Zak Seidov, Aug 31 2015 *)

is_A235110(p, b=13)={my(d=digits(p)); isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)}

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)

A235635 Primes whose base-5 representation is also the base-7 representation of a prime.

Original entry on oeis.org

2, 3, 5, 13, 17, 23, 29, 41, 43, 47, 53, 59, 61, 71, 79, 83, 101, 103, 137, 157, 163, 181, 191, 223, 227, 239, 281, 347, 379, 383, 419, 443, 463, 479, 547, 563, 571, 593, 641, 691, 701, 743, 757, 811, 839, 863, 877, 967, 997, 1049, 1051, 1087, 1097, 1109, 1151, 1171, 1217, 1249, 1259, 1283

Offset: 1

M. F. Hasler, Jan 13 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.

			17 = 32_5 and 32_7 = 23 are both prime, so 17 is a term.

Giovanni Resta, 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. A235627, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235639. See the LINK for further cross-references.

Select[Prime@Range@500, PrimeQ@FromDigits[IntegerDigits[#, 5], 7] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=7,c=5)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A090712 Primes whose base-13 expansion is a (valid) decimal expansion of a prime.

Original entry on oeis.org

2, 3, 5, 7, 29, 53, 59, 79, 107, 113, 241, 263, 269, 293, 367, 373, 419, 443, 521, 523, 547, 601, 607, 631, 677, 757, 761, 787, 937, 971, 1021, 1069, 1093, 1231, 1249, 1277, 1307, 1361, 1381, 1433, 1459, 1543, 1567, 1613, 1619, 2213, 2237, 2239, 2447, 2477

Offset: 1

Cino Hilliard, Jan 18 2004

See A235110 for a similar sequence whose definition works "in the opposite direction": Actually, the base-13 representation of the terms here. - M. F. Hasler, Jan 03 2014

			The prime p = 53 is written 41 in base 13, and 41 is again (the base 10 representation of) a prime. Therefore p = 53 is a term of this sequence. [Rephrased by _M. F. Hasler_, Jan 03 2014]

Robert Price, Table of n, a(n) for n = 1..14473

Cf. A090707 - A091924.

f[n_]:=Module[{c13=FromDigits[IntegerDigits[n],13]},If[PrimeQ[c13], c13,0]]; Select[f/@Prime[Range[500]],#!=0&] (* Harvey P. Dale, Jun 20 2011 *)

is_A090712(p)=vecmax(d=digits(p,13))<10&&isprime(vector(#d,i,10^(#d-i))*d~)&&isprime(p) \\ M. F. Hasler, Jan 05 2014

Edited by N. J. A. Sloane, Feb 07 2007, and by M. F. Hasler, Jan 05 2014

A091923 Primes whose decimal representations interpreted in base 11 are not prime.

Original entry on oeis.org

11, 13, 17, 19, 23, 31, 37, 41, 47, 53, 59, 71, 73, 79, 83, 97, 101, 103, 107, 109, 113, 127, 131, 137, 149, 151, 157, 163, 167, 173, 179, 181, 191, 211, 223, 229, 233, 239, 241, 251, 257, 271, 277, 283, 293, 307, 311, 313, 317, 337, 347, 349, 359, 367, 383

Offset: 1

Reinhard Zumkeller, Feb 13 2004

A090862(A049084(a(n))) = 11.

			A000040(9)=23 in base 11 is 2*11^1 + 3*11^0 = 25 = 5^2, therefore 29 is a term.

Marius A. Burtea, Table of n, a(n) for n = 1..1045

Cf. A091924.

[n:n in PrimesUpTo(400)| not IsPrime(Seqint(Intseq(n), 11))]; // Marius A. Burtea, Jun 30 2019

Select[Prime@Range@80, ! PrimeQ@FromDigits[IntegerDigits@#, 11] &] (* Vincenzo Librandi, Jul 01 2019 *)

isok(p) = isprime(p) && (d=digits(p)) && !isprime(fromdigits(d, 11)); \\ Michel Marcus, Jun 30 2019

Corrected by Zak Seidov, Feb 25 2004

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

A065726 Primes p whose base-8 expansion is also the decimal expansion of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A090862 Smallest b > 10 such that the decimal representation of the n-th prime is not a prime in base b representation.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A235110 Primes whose base-10 representation also represents a prime in base 13.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI