A152079 -id:A152079 - cp's OEIS frontend

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

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.

A235632 Primes whose base-5 representation is also the base-8 representation of a prime.

Original entry on oeis.org

2, 3, 11, 13, 31, 41, 53, 73, 101, 131, 151, 223, 281, 313, 353, 401, 463, 521, 523, 541, 593, 661, 701, 733, 773, 941, 983, 1013, 1063, 1091, 1093, 1123, 1153, 1193, 1201, 1321, 1381, 1423, 1471, 1481, 1483, 1571, 1583, 1601, 1613, 1663, 1693, 1741, 1753, 1801, 1861, 1871, 1873

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_8 = 17 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. A235628, 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[#,5],8]]&] (* Harvey P. Dale, Dec 15 2018 *)

is(p,b=8,c=5)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

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

Original entry on oeis.org

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)

cp's OEIS Frontend

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

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

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

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A235632 Primes whose base-5 representation is also the base-8 representation of a prime.

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

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