A235615 -id:A235615 - cp's OEIS frontend

A235633 Primes whose base-4 representation is also the base-8 representation of a prime.

2, 3, 7, 11, 19, 29, 37, 59, 71, 89, 97, 107, 149, 167, 223, 227, 251, 281, 337, 347, 439, 461, 463, 491, 499, 509, 521, 563, 599, 617, 647, 653, 659, 701, 727, 733, 739, 751, 757, 769, 797, 809, 823, 877, 887, 907, 911, 929, 937, 1031, 1087, 1093, 1109, 1163, 1187, 1193, 1297

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.

			7 = 13_4 and 13_8 = 11 are both prime, so 7 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. A235618, 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[#, 4], 8] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=8,c=4)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A235634 Primes whose base-4 representation is also the base-7 representation of a prime.

Original entry on oeis.org

2, 3, 11, 23, 29, 31, 41, 71, 79, 101, 109, 113, 137, 149, 157, 163, 191, 199, 239, 251, 263, 269, 283, 307, 353, 397, 401, 431, 443, 521, 547, 601, 701, 709, 743, 751, 773, 853, 887, 941, 947, 983, 1013, 1039, 1049, 1069, 1109, 1151, 1217, 1283, 1303, 1487, 1489, 1663, 1669, 1789, 1823, 1901, 1949, 1973

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.

			E.g., 11 = 23_4 and 23_7 = 17 both are prime.

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. A235617, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971, A020449, A089981, A090707 - A091924, A235615 - A235639. See the LINK for further cross-references.

filter:= proc(n) local L,m;
  if not isprime(n) then return false fi;
  L:= convert(n,base,4);
  isprime(add(L[i]*7^(i-1),i=1..nops(L)));
end proc:
select(filter, [2,seq(i,i=3..10000,2)]); # Robert Israel, Jul 02 2018

is(p,b=7,c=4)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A235636 Primes whose base-6 representation is also the base-7 representation of a prime.

Original entry on oeis.org

2, 3, 5, 17, 47, 71, 97, 101, 131, 157, 173, 191, 211, 251, 257, 277, 307, 311, 353, 367, 401, 421, 461, 487, 491, 563, 577, 601, 631, 643, 647, 683, 701, 743, 751, 761, 853, 857, 907, 911, 937, 953, 983, 1021, 1033, 1087, 1103, 1193, 1201, 1259, 1277, 1289, 1327, 1451, 1471, 1571, 1583, 1597, 1601, 1747, 1831, 1907, 1933

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 = 25_6 and 25_7 = 19 are both prime.

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. A235637, 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[300]],PrimeQ[FromDigits[IntegerDigits[#,6],7]]&] (* Harvey P. Dale, Sep 17 2017 *)

is(p,b=7,c=6)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A235637 Primes whose base-7 representation also is the base-6 representation of a prime.

Original entry on oeis.org

2, 3, 5, 19, 61, 89, 127, 131, 173, 211, 229, 257, 281, 383, 397, 421, 463, 467, 523, 547, 593, 617, 719, 757, 761, 859, 883, 911, 953, 967, 971, 1069, 1097, 1153, 1163, 1181, 1303, 1307, 1429, 1433, 1471, 1489, 1531, 1583, 1597, 1723, 1741, 1867, 1877, 1951, 1979, 1993, 2437

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.

			E.g., 19 = 25_7 and 25_6 = 17 are both prime.

Robert Price, Table of n, a(n) for n = 1..14278
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235636, 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=6,c=7)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,7,6)&&print1(vector(#d=digits(p,6),i,7^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,6,7)

A235619 Primes whose base-9 representation also is the base-4 representation of a prime.

Original entry on oeis.org

2, 3, 109, 181, 271, 829, 919, 1549, 1567, 6661, 6733, 6823, 8101, 8191, 8263, 13933, 14851, 15319, 22123, 59149, 59221, 59239, 59797, 59887, 61507, 65629, 65701, 72253, 72901, 73819, 118189, 118927, 119827, 124669, 125407, 126127, 126307, 132679, 132859

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.

			E.g., 109 = 131_9 and 131_4 = 29 are both prime.

Robert Price, Table of n, a(n) for n = 1..1296
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235481, 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=9)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,9,4)&&print1(vector(#d=digits(p,4),i,9^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,4,9)

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

Original entry on oeis.org

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)

A359840 Numbers k that are the representation of primes in base 4 and in base 5.

Original entry on oeis.org

2, 3, 23, 131, 133, 221, 1211, 1231, 2023, 2111, 2113, 2311, 3013, 3211, 3233, 3323, 10031, 10033, 10121, 12011, 12121, 13223, 13331, 20131, 20203, 22111, 23233, 31313, 32033, 32303, 33133, 33331, 100123, 100211, 100231, 101003, 101333, 103333, 110021, 111211

Offset: 1

Bernard Schott, Jan 15 2023

For a(1) = 2, 2_4 = 2_5 = 2_10 and for a(2) = 3, 3_4 = 3_5 = 3_10; otherwise, these two primes are distinct for n >= 3 (example).

The corresponding sequences of primes are A235474 (for base 4) and A235615 (for base 5).

			a(3) = 23 because 23_4 = 11_10 = A235474(3) and 23_5 = 13_10 = A235615(3) are primes.
a(9) = 2023 because 2023_4 = 139_10 = A235474(9) and 2023_5 = 263_10 = A235615(9) are primes.

Intersection of A004678 and A004679.

Cf. A007090, A007091, A235474, A235615, A340290.

q[n_, b_] := Max[d = IntegerDigits[n]] < b && PrimeQ[FromDigits[d, b]]; Select[Range[200000], q[#, 4] && q[#, 5] &] (* Amiram Eldar, Jan 15 2023 *)

from sympy import isprime
def ok(n): return max(s:=str(n)) < '4' and isprime(int(s, 4)) and isprime(int(s, 5))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jan 15 2023

from sympy import isprime
from itertools import count, islice, product
def agen(): yield from (int(s) for d in count(1) for f in "123" for r in product("0123", repeat=d-1) if isprime(int(s:=f+"".join(r), 4)) and isprime(int(s, 5)))
print(list(islice(agen(), 40))) # Michael S. Branicky, Jan 15 2023

a(n) = A007090(A235474(n)); a(n) = A007091(A235615(n)).

More terms from Amiram Eldar, Jan 15 2023

cp's OEIS Frontend

A235633 Primes whose base-4 representation is also the base-8 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A235634 Primes whose base-4 representation is also the base-7 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A235636 Primes whose base-6 representation is also the base-7 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A235637 Primes whose base-7 representation also is the base-6 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A235619 Primes whose base-9 representation also is the base-4 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A359840 Numbers k that are the representation of primes in base 4 and in base 5.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Python

Formula