A065721 -id:A065721 - cp's OEIS frontend

A231479 Primes whose base-7 representation is also the base-9 representation of a prime.

2, 3, 5, 11, 19, 23, 29, 37, 47, 67, 71, 89, 103, 107, 113, 127, 137, 163, 179, 239, 257, 313, 337, 347, 389, 401, 431, 457, 463, 499, 523, 547, 569, 571, 617, 709, 719, 739, 743, 751, 757, 761, 821, 823, 859, 883, 887, 971, 1019, 1069, 1093, 1129, 1153, 1213, 1297, 1307, 1327, 1367, 1373, 1381

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.

			11 = 14_7 and 14_9 = 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. A235621, A235265, A235266, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924. See the LINK for further cross-references.

Select[Prime@ Range@ 250, PrimeQ@ FromDigits[IntegerDigits[#, 7], 9] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=9,c=7)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A235463 Primes whose base-6 representation also is the base-2 representation of a prime.

Original entry on oeis.org

7, 37, 43, 223, 1297, 1303, 1549, 7993, 9109, 46663, 54469, 55987, 326593, 1679659, 1681129, 1727569, 1734049, 1967587, 2006461, 2007763, 2014027, 2015287, 10077919, 10125649, 10125691, 10133467, 10412107, 10413397, 11757349, 11766421, 11766427, 11766637

Offset: 1

M. F. Hasler, Jan 11 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.
When the smaller base is b=2 such that only digits 0 and 1 are allowed, these are primes that are the sum of distinct powers of the larger base, here c=6, thus a subsequence of A077720.

			7 = 11_6 and 11_2 = 3 are both prime, so 7 is a term.
37 = 101_6 and 101_2 = 5 are both prime, so 37 is a term.

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

  b62Q[n_]:=Module[{idn6=IntegerDigits[n,6]},Max[idn6]<2&&AllTrue[ {FromDigits[ idn6,6],FromDigits[idn6,2]},PrimeQ]]; Select[Prime[ Range[ 4,780000]],b62Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 29 2020 *)

is(p,b=2,c=6)=vecmax(d=digits(p,c))

forprime(p=1,1e3,is(p,6,2)&&print1(vector(#d=digits(p,2),i,6^(#d-i))*d~,",")) \\ To produce the terms, this is much more efficient than to select them using straightforwardly is(.)=is(.,2,6)

A235465 Primes whose base-8 representation also is the base-2 representation of a prime.

Original entry on oeis.org

73, 521, 577, 4673, 32833, 33289, 33353, 36929, 37441, 262153, 262217, 262657, 295433, 2097673, 2101313, 2359369, 2363401, 2392073, 16777289, 16810049, 16814089, 16814153, 16814657, 17039881, 17043977, 17076809, 18874433, 18907201, 19137089, 19140617, 134222401, 134483969, 134484481, 134513161

Offset: 1

M. F. Hasler, Jan 11 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.
When the smaller base is b=2 such that only digits 0 and 1 are allowed, these are primes that are the sum of distinct powers of the larger base, here c=8, thus a subsequence of A077722.

			73 = 111_8 and 111_2 = 7 are both prime, so 73 is a term.

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. A235478, A235479, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

is(p,b=2,c=8)=vecmax(d=digits(p,c))

forprime(p=1,1e3,is(p,8,2)&&print1(vector(#d=digits(p,2),i,8^(#d-i))*d~,",")) \\ To produce the terms, this is much more efficient than to select them using straightforwardly is(.)=is(.,2,8)

A235466 Primes whose base-9 representation also is the base-2 representation of a prime.

Original entry on oeis.org

739, 811, 6571, 59779, 532261, 591301, 4783699, 4789621, 4842109, 4849399, 5314411, 5314501, 5373469, 5374279, 43047541, 43112341, 43113061, 47888821, 47889559, 47895301, 48361861, 48420271, 48420919, 387421219, 387486109, 388011061, 388011709, 392210029, 392262589, 392734981

Offset: 1

M. F. Hasler, Jan 11 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.
When the smaller base is b=2 such that only digits 0 and 1 are allowed, these are primes that are the sum of distinct powers of the larger base, here c=9, thus a subsequence of A077723.

			739 = 1011_9 and 1011_2 = 11 are both prime, so 739 is a term.

Zak Seidov, Table of n, a(n) for n = 1..1400
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.

fQ[n_, j_, k_] := Block[{id = IntegerDigits[n, j]}, Max[id] < k && PrimeQ[ FromDigits[ id, k]]]; lst = {}; p = 2; While[p < 4*10^9, If[ fQ[p, 9, 2], AppendTo[lst, p]; Print[p]]; p = NextPrime@ p] (* Robert G. Wilson v, Oct 09 2014 *)
pr9Q[n_]:=Module[{idn9=IntegerDigits[n,9]},Max[idn9]<2&&PrimeQ[ FromDigits[ idn9,2]]]; Select[Prime[Range[21*10^6]],pr9Q] (* Harvey P. Dale, Aug 25 2015 *)

is(p,b=2,c=9)=vecmax(d=digits(p,c))

forprime(p=1,1e3,is(p,9,2)&&print1(vector(#d=digits(p,2),i,9^(#d-i))*d~,",")) \\ To produce the terms, this is much more efficient than to select them using straightforwardly is(.)=is(.,2,9)

A235467 Primes whose base-4 representation also is the base-3 representation of a prime.

Original entry on oeis.org

2, 89, 137, 149, 281, 293, 353, 389, 409, 421, 593, 613, 661, 1097, 1109, 1289, 1301, 1321, 1381, 1409, 1601, 1609, 1669, 2069, 2129, 2309, 2377, 2389, 2729, 4133, 4229, 4373, 4441, 4513, 4673, 5153

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.
This is a subsequence of A002144, A002313, A003655, A050150, A062090, A141293, A175768, A192592, A226181 (conjectural).

			E.g., 89 = 1121_4 and 1121_3 = 43 both are 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. A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

b4b3Q[n_]:=Module[{b4=IntegerDigits[n,4]},Max[b4]<3&&PrimeQ[ FromDigits[ b4,3]]]; Select[Prime[Range[700]],b4b3Q] (* Harvey P. Dale, Dec 14 2021 *)

is(p,b=3,c=4)=vecmax(d=digits(p,c))

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

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

Original entry on oeis.org

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.

A267765 Numbers whose base-5 representation is a square when read in base 10.

Original entry on oeis.org

0, 1, 4, 25, 36, 49, 89, 100, 121, 139, 249, 329, 351, 625, 676, 729, 900, 961, 1225, 1551, 1654, 2146, 2225, 2289, 2500, 2601, 3025, 3289, 3475, 3521, 3814, 4324, 4529, 4801, 5086, 5149, 6225, 6726, 6829, 7374, 8225, 8464, 8775, 9454, 9601, 13926, 15625, 15876, 16129, 16900, 17161

Offset: 1

M. F. Hasler, Jan 20 2016

Trivially includes powers of 25, since 25^k = 100..00_5 = 10^(2k) when read in base 10. Moreover, for any a(n) in the sequence, 25*a(n) is also in the sequence. One could call "primitive" the terms not of this form, these would be 1, 4, 36 = 121_5, 49 = 144_5, 89 = 324_5, ... These primitive terms include the subsequence 25^k + 2*5^k + 1 = (5^k+1)^2, k > 0, which yields A033934 when written in base 5.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A267763 - A267769 for bases 3 through 9. The base-2 analog is A000302 = powers of 4.

For a "prime" analog see also A235265, A235266, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924.

Select[Range[0, 17200], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 5] &] (* Michael De Vlieger, Jan 24 2016 *)

is(n,b=5,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

A267765_list = [int(d,5) for d in (str(i**2) for i in range(10**6)) if max(d) < '5'] # Chai Wah Wu, Mar 12 2016

cp's OEIS Frontend

A231479 Primes whose base-7 representation is also the base-9 representation of a prime.

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A235463 Primes whose base-6 representation also is the base-2 representation of a prime.

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A235465 Primes whose base-8 representation also is the base-2 representation of a prime.

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A235466 Primes whose base-9 representation also is the base-2 representation of a prime.

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A235467 Primes whose base-4 representation also is the base-3 representation of a prime.

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

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

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