A036952 -id:A036952 - cp's OEIS frontend

A036956 Primes containing only digits from the set (0,1,2,3,4).

2, 3, 11, 13, 23, 31, 41, 43, 101, 103, 113, 131, 211, 223, 233, 241, 311, 313, 331, 401, 421, 431, 433, 443, 1013, 1021, 1031, 1033, 1103, 1123, 1201, 1213, 1223, 1231, 1301, 1303, 1321, 1423, 1433, 2003, 2011, 2111, 2113, 2131, 2141, 2143, 2203, 2213

Offset: 1

Patrick De Geest, Jan 04 1999

Jason Bard, Table of n, a(n) for n = 1..10000
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).

Subsequence of A036958.

Cf. A036952-A036964.

Offet 1 from Michel Marcus, Oct 10 2019

A235479 Primes whose base-2 representation also is the base-9 representation of a prime.

Original entry on oeis.org

11, 13, 19, 41, 79, 109, 137, 151, 167, 191, 193, 199, 227, 239, 271, 307, 313, 421, 431, 433, 457, 487, 491, 521, 563, 613, 617, 659, 677, 709, 727, 757, 929, 947, 1009, 1033, 1051, 1249, 1483, 1693, 1697, 1709, 1721, 1831, 1951, 1979, 1987, 1993

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.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
A subsequence of A027697, A050150, A062090 and A176620.

			11 = 1011_2 and 1011_9 = 6571 are both prime, so 11 is a term.

Robert Price, Table of n, a(n) for n = 1..1958
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235466 ⊂ A077723, A235266, A152079, A235475 - A235478, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

is(p,b=9)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

A267490 Primes whose base-8 representation is a perfect square in base 10.

Original entry on oeis.org

149, 241, 661, 1409, 2593, 3733, 6257, 7793, 15313, 23189, 25601, 26113, 30497, 34337, 44053, 49057, 78577, 92821, 95009, 108529, 115861, 132757, 162257, 178417, 183377, 223381, 235541, 242197, 266261, 327317, 345749, 426389, 525461, 693397, 719893, 729713, 805397, 814081, 903841

Offset: 1

Christopher Cormier, Jan 16 2016

Subsequence of primes in A267768. - M. F. Hasler, Jan 20 2016

			a(1) = 149 because 149 is 225 in base 8, and 225 is 15^2 in base 10.

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

For primes which are primes in other bases, see A235265, A235266, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924.

[n:n in PrimesUpTo(1000000)| IsSquare(Seqint(Intseq(n,8)))]; // Marius A. Burtea, Jun 30 2019

Select[Prime@ Range[10^5], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 8] &] (* Michael De Vlieger, Jan 16 2016 *)

listp(nn) = {forprime(p=1, nn, d = digits(p, 8); pd = Pol(d); if (issquare(subst(pd, x, 10)), print1(p, ", ")););} \\ Michel Marcus, Jan 16 2016

is(n,b=8,c=10)={issquare(subst(Pol(digits(n,b)),x,c))&&isprime(n)} \\ M. F. Hasler, Jan 20 2016

from sympy import isprime
A267490_list = [int(s,8) for s in (str(i**2) for i in range(10**6)) if max(s) < '8' and isprime(int(s,8))] # Chai Wah Wu, Jan 20 2016

A036954 Primes with digits in {0,1,2} taken as base 3 and converted to base 10.

Original entry on oeis.org

2, 4, 10, 22, 34, 46, 58, 67, 79, 94, 103, 139, 145, 157, 166, 169, 172, 181, 190, 193, 199, 205, 211, 214, 229, 277, 283, 295, 298, 307, 313, 349, 367, 373, 391, 394, 409, 421, 433, 439, 463, 466, 478, 505, 517, 523, 529, 535, 541, 547, 556, 559, 571, 577

Offset: 1

Patrick De Geest, Jan 04 1999

Equivalently: terms of A036953 read in base 3 (and written in base 10). - M. F. Hasler, Jul 25 2015

Equivalently, k such that A007089(k), read literally as a decimal number, is a prime. - N. J. A. Sloane, Feb 17 2023

			a(n) = 313 is 102121{3}, and 102121{10} is prime.

Cf. A036952-A036964.

Indices of primes in A007089.

FromDigits[#,3]&/@Select[Tuples[{0,1,2},6],PrimeQ[FromDigits[#]]&] (* Harvey P. Dale, Mar 27 2021 *)

is(n)=(n%3==1||n==2)&&isprime((n=digits(n,3))*vectorv(#n,i,10^(#n-i))) \\ M. F. Hasler, Jul 25 2015

a(n) == 1 (mod 3) for all n > 1. - M. F. Hasler, Jul 25 2015

Offset corrected to 1 and minor edits by M. F. Hasler, Jul 25 2015

A036958 Primes containing only digits from the set (0,1,2,3,4,5).

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 31, 41, 43, 53, 101, 103, 113, 131, 151, 211, 223, 233, 241, 251, 311, 313, 331, 353, 401, 421, 431, 433, 443, 503, 521, 523, 541, 1013, 1021, 1031, 1033, 1051, 1103, 1123, 1151, 1153, 1201, 1213, 1223, 1231, 1301, 1303, 1321

Offset: 1

Patrick De Geest, Jan 04 1999

Harvey P. Dale, Table of n, a(n) for n = 1..1000 [offset shifted by _Georg Fischer_, Oct 14 2019]
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)

Subsequence of A036960.

Cf. A036952-A036964.

Select[Prime[Range[220]],Max[IntegerDigits[#]]<6&] (* Harvey P. Dale, Sep 23 2011 *)

Offset 1 from Michel Marcus, Oct 10 2019

A235474 Primes whose base-4 representation is also the base-5 representation of a prime.

Original entry on oeis.org

2, 3, 11, 29, 31, 41, 101, 109, 139, 149, 151, 181, 199, 229, 239, 251, 269, 271, 281, 389, 409, 491, 509, 541, 547, 661, 751, 887, 911, 947, 991, 1021, 1051, 1061, 1069, 1091, 1151, 1279, 1289, 1381, 1409, 1471, 1549, 1709, 1759, 1801, 1999

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.

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

			11 = 23_4 and 23_5 = 13 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. A235266, A235473, A152079, A235475 - A235479, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,4],5]]&] (* Harvey P. Dale, Dec 31 2017 *)

is(p,b=5,c=4)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A235477 Primes whose base-2 representation also is the base-7 representation of a prime.

Original entry on oeis.org

2, 31, 47, 59, 103, 107, 173, 179, 181, 199, 211, 227, 229, 233, 367, 409, 443, 463, 487, 701, 743, 757, 823, 827, 877, 911, 919, 967, 1009, 1123, 1163, 1291, 1321, 1367, 1373, 1447, 1493, 1571, 1583, 1597, 1609, 1627, 1657, 1669, 1721, 1831, 1933, 1987

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.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
A subsequence of A027697, A015919, A197636 (conjectural).

			31 = 11111_2 and 11111_7 = 2801 are both prime, so 31 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. A235464 ⊂ A077721, A235475, A152079, A235266, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[300]],PrimeQ[FromDigits[IntegerDigits[#,2],7]]&] (* Harvey P. Dale, May 08 2021 *)

is(p,b=7)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

A036960 Primes containing only digits from the set (0,1,2,3,4,5,6).

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 31, 41, 43, 53, 61, 101, 103, 113, 131, 151, 163, 211, 223, 233, 241, 251, 263, 311, 313, 331, 353, 401, 421, 431, 433, 443, 461, 463, 503, 521, 523, 541, 563, 601, 613, 631, 641, 643, 653, 661, 1013, 1021, 1031, 1033, 1051, 1061, 1063

Offset: 1

Patrick De Geest, Jan 04 1999

James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)

Subsequence of A036962.

Cf. A036952-A036964.

Select[Prime[Range[200]],Max[IntegerDigits[#]]<7&] (* Harvey P. Dale, Feb 08 2019 *)

Offset 1 from Michel Marcus, Oct 10 2019

A231474 Primes whose base-3 representation is also the base-5 representation of a prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 29, 31, 37, 41, 59, 67, 79, 97, 101, 109, 113, 137, 139, 149, 151, 173, 181, 193, 223, 229, 251, 269, 271, 293, 311, 331, 353, 367, 373, 379, 383, 389, 397, 401, 457, 467, 491, 503, 617, 631, 641, 647, 653, 673, 701, 773, 787, 797, 809, 829, 853, 857, 911, 929, 953, 977

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.

			7 = 21_3 and 21_5 = 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. A235265, A235266, A235473, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924. See the LINK for further cross-references.

Select[Prime@ Range@ 500, PrimeQ@ FromDigits[ IntegerDigits[#, 3], 5] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=5,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A231477 Primes whose base-3 representation is also the base-7 representation of a prime.

Original entry on oeis.org

2, 3, 23, 41, 47, 53, 61, 67, 71, 89, 113, 127, 131, 137, 191, 193, 223, 251, 269, 283, 293, 311, 353, 397, 409, 421, 443, 463, 491, 503, 509, 541, 569, 601, 613, 701, 773, 787, 983, 1013, 1031, 1091, 1117, 1213, 1223, 1429, 1499, 1543, 1549, 1579, 1619, 1621, 1697, 1699, 1733, 1873, 1933, 1949, 1951, 1973

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.

			23 = 212_3 and 212_7 = 107 are both prime, so 23 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. A235470, 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@500, PrimeQ@FromDigits[IntegerDigits[#, 3], 7] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=7,c=3)=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

A036956 Primes containing only digits from the set (0,1,2,3,4).

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A235479 Primes whose base-2 representation also is the base-9 representation of a prime.

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A267490 Primes whose base-8 representation is a perfect square in base 10.

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A036954 Primes with digits in {0,1,2} taken as base 3 and converted to base 10.

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A036958 Primes containing only digits from the set (0,1,2,3,4,5).

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A235474 Primes whose base-4 representation is also the base-5 representation of a prime.

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A235477 Primes whose base-2 representation also is the base-7 representation of a prime.

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Mathematica

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A036960 Primes containing only digits from the set (0,1,2,3,4,5,6).

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