A235265 -id:A235265 - cp's OEIS frontend

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

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.

A235625 Primes whose base-5 representation is also the base-6 representation of a prime.

Original entry on oeis.org

2, 3, 11, 31, 71, 131, 191, 211, 241, 251, 271, 331, 421, 431, 461, 491, 541, 601, 631, 811, 821, 911, 971, 1031, 1051, 1061, 1171, 1181, 1201, 1231, 1291, 1321, 1361, 1451, 1511, 1531, 1571, 1601, 1721, 1801, 1811, 1831, 1861, 1931, 2081, 2111, 2131, 2141, 2311, 2341, 2381, 2411, 2521, 2531, 2711, 2741, 2801

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.

			11 = 21_5 and 21_6 = 13 are both prime, so 11 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. A235626, 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], 6] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=6,c=5)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A235626 Primes whose base-6 representation also is the base-5 representation of a prime.

Original entry on oeis.org

2, 3, 13, 43, 97, 223, 307, 337, 379, 433, 457, 547, 709, 727, 769, 811, 919, 1009, 1303, 1597, 1609, 1777, 1861, 1987, 2017, 2029, 2221, 2239, 2269, 2311, 2647, 2689, 2749, 2917, 3037, 3067, 3121, 3169, 3373, 3529, 3541, 3571, 3613, 3967, 4219, 4261, 4327, 4339

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 = 21_6 and 21_5 = 11 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. A235625, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235639. See the LINK for further cross-references.

P:= {seq(ithprime(i),i=1..10000)}:
f:= proc(p) local i,L;
  L:= convert(p,base,5);
  add(L[i]*6^(i-1),i=1..nops(L))
end proc:
sort(convert(map(f,P) intersect P,list)); # Robert Israel, Jun 18 2019

b65pQ[n_]:=Module[{idn6=IntegerDigits[n,6]},Max[idn6]<5&&PrimeQ[ FromDigits[ idn6,5]]]; Select[Prime[Range[600]],b65pQ] (* Harvey P. Dale, Oct 13 2020 *)

is(p,b=5,c=6)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,6,5)&&print1(vector(#d=digits(p,5),i,6^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,4,6)

A235627 Primes whose base-7 representation also is the base-5 representation of a prime.

Original entry on oeis.org

2, 3, 7, 17, 23, 31, 53, 71, 73, 79, 101, 109, 113, 127, 151, 157, 197, 199, 359, 401, 409, 449, 463, 521, 541, 557, 743, 863, 1033, 1039, 1103, 1151, 1193, 1229, 1451, 1487, 1499, 1543, 2423, 2521, 2549, 2621, 2753, 2857, 2909, 2957, 3089, 3257, 3313, 3511, 3529, 3593

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 17 = 23_7 and 23_5 = 13 are prime.

Robert Price, Table of n, a(n) for n = 1..13736
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235635, 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=5,c=7)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,7,5)&&print1(vector(#d=digits(p,5),i,7^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,5,7)

A235628 Primes whose base-8 representation also is the base-5 representation of a prime.

Original entry on oeis.org

2, 3, 17, 19, 73, 89, 131, 163, 257, 521, 577, 739, 1097, 1171, 1283, 1601, 1747, 2081, 2083, 2137, 2267, 4177, 4289, 4363, 4643, 5273, 5387, 5651, 5779, 5849, 5851, 5923, 6211, 6299, 6337, 8353, 8713, 8803, 8929, 8969, 8971, 9377, 9419, 9473, 9491, 9811, 9883, 10009

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 17 = 23_8 and 23_5 = 13 are prime.

Robert Price, Table of n, a(n) for n = 1..3395
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235632, 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=5,c=8)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,8,5)&&print1(vector(#d=digits(p,5),i,8^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,5,8)

A235630 Primes whose base-7 representation is also the base-8 representation of a prime.

Original entry on oeis.org

2, 3, 5, 17, 47, 71, 89, 101, 197, 229, 241, 269, 271, 337, 353, 383, 479, 521, 577, 607, 631, 647, 673, 677, 719, 743, 761, 827, 997, 1097, 1153, 1181, 1193, 1279, 1289, 1303, 1319, 1447, 1543, 1601, 1697, 1811, 1823, 1907, 1951, 1993, 2017, 2131, 2203, 2243, 2339, 2357, 2383, 2549

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 = 23_7 and 23_8 = 19 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. A235622, 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[#, 7], 8] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=8,c=7)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A235631 Primes whose base-6 representation is also the base-8 representation of a prime.

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 29, 31, 43, 61, 71, 79, 89, 107, 109, 113, 137, 139, 163, 173, 193, 223, 239, 251, 271, 281, 283, 313, 317, 347, 383, 431, 439, 461, 467, 491, 499, 541, 557, 593, 607, 641, 659, 661, 691, 701, 743, 761, 853, 863, 881, 919, 971, 997, 1013, 1031, 1051, 1061, 1063

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.

			11 = 15_6 and 15_8 = 13 are both prime, so 11 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. A235638, 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[#, 6], 8] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=8,c=6)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

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

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

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

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

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A235625 Primes whose base-5 representation is also the base-6 representation of a prime.

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A235626 Primes whose base-6 representation also is the base-5 representation of a prime.

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A235627 Primes whose base-7 representation also is the base-5 representation of a prime.

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