A065727 -id:A065727 - cp's OEIS frontend

A235469 Primes whose base-6 representation also is the base-3 representation of a prime.

2, 13, 43, 73, 223, 1777, 2593, 2887, 3037, 3067, 3109, 7993, 9157, 9337, 10597, 17077, 17107, 17137, 17317, 17359, 18229, 18661, 46663, 48247, 49297, 49537, 54517, 54727, 54877, 54907, 54949, 55987, 56197, 56209, 56239, 57097, 63589, 63727, 64879, 65089, 65101, 95089, 95917, 96157

Offset: 1

M. F. Hasler, Jan 12 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.

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

			E.g., 13 = 21_6 and 21_3 = 7 are both prime.

Robert Price, Table of n, a(n) for n = 1..1959
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

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

is(p,b=3,c=6)=vecmax(d=digits(p,c))

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

A235470 Primes whose base-7 representation also is the base-3 representation of a prime.

Original entry on oeis.org

2, 7, 107, 401, 443, 457, 701, 743, 751, 2417, 2753, 2843, 2851, 3089, 5147, 5153, 5503, 16823, 16921, 17207, 17257, 17551, 19553, 19993, 21617, 21673, 22003, 22303, 33623, 33679, 33721, 34301, 36017, 36373, 36457, 38873, 118057, 118343, 134507, 134857, 135151, 137251, 137593, 140057

Offset: 1

M. F. Hasler, Jan 12 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.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
Since the trailing digit of the base-7 expansion must (like all others) be less than 3, this is a subsequence of A045381.

			E.g., 7 = 10_7 and 10_3 = 3 are both prime; 107 = 212_7 and 212_3 = 23 are both prime.

Robert Price, Table of n, a(n) for n = 1..829
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

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

is(p,b=3,c=7)=vecmax(d=digits(p,c))

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

A235471 Primes whose base-8 representation also is the base-3 representation of a prime.

Original entry on oeis.org

2, 17, 73, 521, 577, 593, 1097, 1153, 4177, 8713, 33353, 33857, 37889, 41617, 65537, 65609, 69697, 70289, 70793, 74897, 262153, 262657, 266369, 331777, 331921, 336529, 336977, 529489, 533129, 533633, 590921, 594953, 598537, 2098241, 2101249, 2102417, 2134529

Offset: 1

M. F. Hasler, Jan 12 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.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
Seems to be a subsequence of A066649 and A123364.
Since the trailing digit of the base 7 expansion must (like all others) be less than 3, this is a subsequence of A045381.

			E.g., 17 = 21_8 and 21_3 = 7 are both 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. A231478, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482, A235615 - A235639. See the LINK for further cross-references.

b8b3pQ[n_]:=Module[{id8=IntegerDigits[n,8]},Max[id8]<3&&PrimeQ[ FromDigits[ id8,3]]]; Select[Prime[Range[160000]],b8b3pQ] (* Harvey P. Dale, Mar 16 2019 *)

is(p,b=3,c=8)=vecmax(d=digits(p,c))

forprime(p=1,1e3,is(p,8,3)&&print1(vector(#d=digits(p,3),i,8^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,3,8)

A235480 Primes whose base-3 representation is also the base-9 representation of a prime.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 23, 31, 37, 41, 43, 53, 67, 71, 73, 83, 89, 97, 103, 149, 157, 199, 239, 251, 257, 271, 277, 293, 307, 313, 331, 337, 359, 383, 397, 421, 431, 433, 499, 541, 557, 571, 587, 599, 601, 613, 631, 653, 659, 661, 683, 691, 709, 727, 751, 769, 823, 887, 911, 983, 1009, 1021, 1031, 1049, 1051, 1063, 1129, 1163, 1217

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.

Appears to be a subsequence of A015919, A045344, A052085, A064555 and A143578.

			5 = 12_3 and 12_9 = 11 are both prime, so 5 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. A235265, A235473 - A235479, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime@ Range@ 500, PrimeQ@ FromDigits[ IntegerDigits[#, 3], 9] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=9,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: Code only valid for b > c.

A235617 Primes whose base-7 representation also is the base-4 representation of a prime.

Original entry on oeis.org

2, 3, 17, 59, 71, 73, 113, 353, 367, 449, 463, 491, 701, 743, 757, 787, 857, 1039, 1151, 1193, 2411, 2423, 2467, 2551, 2843, 3109, 3137, 3209, 3251, 4817, 4903, 5209, 5657, 5839, 5939, 5953, 7211, 7603, 7703, 8009, 8039, 8291, 8387, 16831, 16871, 16927, 17207, 17321, 17837, 19211, 19267, 20261, 20287, 22123, 22303

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

Robert Price, Table of n, a(n) for n = 1..8515
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235634, 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=7)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,7,4)&&print1(vector(#d=digits(p,4),i,7^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,4,9)

A235618 Primes whose base-8 representation also is the base-4 representation of a prime.

Original entry on oeis.org

2, 3, 11, 19, 67, 89, 137, 211, 523, 593, 641, 659, 1097, 1163, 1627, 1667, 1747, 4177, 4673, 4691, 5323, 5657, 5659, 5779, 5827, 5849, 8209, 8387, 8779, 8849, 9227, 9241, 9283, 9433, 9803, 9817, 9859, 9883, 9929, 12289, 12377, 12433, 12491, 12953, 13003, 13331, 13339, 13441

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., 11 = 13_8 and 13_4 = 7 are both prime.

Robert Price, Table of n, a(n) for n = 1..1627
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235633, 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=8)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,8,4)&&print1(vector(#d=digits(p,4),i,8^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,4,9)

A235620 Primes whose base-9 representation also is the base-8 representation of a prime.

Original entry on oeis.org

2, 3, 5, 7, 19, 41, 59, 97, 109, 131, 151, 277, 331, 347, 457, 491, 541, 547, 577, 601, 739, 761, 811, 829, 977, 997, 1031, 1231, 1279, 1303, 1321, 1499, 1549, 1571, 1609, 1621, 1801, 1987, 2221, 2239, 2269, 2309, 2381, 2399, 2521, 2617, 2687, 2707, 2791, 2939, 2953, 3119

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 is a term: 19 = 21_9 and 21_8 = 17, also a prime.
79 is not a term: 79 = 87_9 and 87 is not a valid base-8 representation.

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. A231480, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235639. See the LINK for further cross-references.

b9b8pQ[n_]:=Module[{id=IntegerDigits[n,9]},Max[id]<8&&PrimeQ[FromDigits[ id,8]]]; Select[Prime[Range[500]],b9b8pQ] (* Harvey P. Dale, Mar 12 2018 *)

is(p,b=8,c=9)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,9,8)&&print1(vector(#d=digits(p,8),i,9^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,8,9)

isok(p) = isprime(p) && (q = digits(p, 9)) && (vecmax(q) < 8) && isprime(fromdigits(q, 8)); \\ Michel Marcus, Mar 12 2018

A235621 Primes whose base-9 representation also is the base-7 representation of a prime.

Original entry on oeis.org

2, 3, 5, 13, 23, 29, 37, 47, 59, 103, 109, 131, 167, 173, 181, 199, 211, 263, 283, 379, 419, 509, 541, 733, 787, 821, 859, 911, 919, 983, 1013, 1063, 1091, 1093, 1171, 1487, 1499, 1543, 1549, 1559, 1567, 1571, 1667, 1669, 1733, 1783, 1787, 1913, 1993, 2237, 2287, 2351, 2381, 2477, 2621

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., 13 = 14_9 and 14_7 = 11 are both prime.

Robert Price, Table of n, a(n) for n = 1..14672
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A231479, 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=7,c=9)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,9,7)&&print1(vector(#d=digits(p,7),i,9^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,7,9)

A235622 Primes whose base-8 representation also is the base-7 representation of a prime.

Original entry on oeis.org

2, 3, 5, 19, 53, 89, 109, 131, 257, 293, 307, 347, 349, 433, 523, 557, 683, 739, 811, 853, 881, 907, 937, 941, 1061, 1097, 1117, 1201, 1427, 1621, 1693, 1733, 1747, 1861, 1873, 1889, 1907, 2141, 2267, 2341, 2467, 2677, 2699, 2803, 2861, 2917, 2953, 3163, 3253, 3307, 3433

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 = 23_8 and 23_7 = 17 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. A235630, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235639. See the LINK for further cross-references.

pb87Q[n_]:=Module[{idn8=IntegerDigits[n,8]},Max[idn8]<7&&PrimeQ[ FromDigits[ idn8,7]]]; Select[Prime[Range[500]],pb87Q] (* Harvey P. Dale, Dec 13 2016 *)

is(p,b=7,c=8)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,8,7)&&print1(vector(#d=digits(p,7),i,8^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,7,8)

A235624 Primes whose base-4 representation is also the base-6 representation of a prime.

Original entry on oeis.org

2, 3, 5, 13, 17, 37, 61, 73, 109, 157, 173, 181, 229, 233, 241, 257, 317, 337, 349, 373, 397, 409, 541, 557, 569, 601, 613, 661, 761, 769, 797, 821, 857, 953, 1013, 1021, 1033, 1069, 1153, 1181, 1193, 1201, 1229, 1237, 1297, 1321, 1373, 1429, 1481, 1609, 1621, 1637, 1709, 1801, 1861, 1877, 1889, 1901, 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.

			5 = 11_4 and 11_6 = 7 are both prime, so 5 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. A235616, 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], 6] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=6,c=4)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

cp's OEIS Frontend

A235469 Primes whose base-6 representation also is the base-3 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A235470 Primes whose base-7 representation also is the base-3 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A235471 Primes whose base-8 representation also is the base-3 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A235480 Primes whose base-3 representation is also the base-9 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A235617 Primes whose base-7 representation also is the base-4 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A235618 Primes whose base-8 representation also is the base-4 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A235620 Primes whose base-9 representation also is the base-8 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI