A235265 -id:A235265 - cp's OEIS frontend

A235383 Fibonacci numbers that are the product of other Fibonacci numbers.

8, 144

Offset: 1

Robert C. Lyons, Jan 08 2014

This sequence and A229037 and A235265 are winners in the contest held at the 2014 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 20 2014
Carmichael's theorem implies that 8 and 144 are the only terms of this sequence.
First two terms of A061899, A111687, A172150, A212703, and A231851. - Omar E. Pol, Jan 21 2014
Saha and Karthik conjectured (without reference to Carmichael's theorem) that the only positive integers k for which A001175(k^2) = A001175(k) are 6 and 12. (A000045(6) = 8 and A000045(12) = 144.) - L. Edson Jeffery, Feb 13 2014
Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only nontrivial perfect power Fibonacci numbers. - Robert C. Lyons, Dec 23 2015

			The Fibonacci number 8 is in the sequence because 8=2*2*2, and 2 is a Fibonacci number that is not equal to 8. The Fibonacci number 144 is in the sequence because 144=3*3*2*2*2*2, and both 2 and 3 are Fibonacci numbers that are not equal to 144.

Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Annals of Mathematics, 163 (2006), pp. 969-1018.
Arpan Saha and Karthik C S, A few equivalences of Wall-Sun-Sun prime conjecture, arXiv:1102.1636 [math.NT], 2011.
Wikipedia, Carmichael's theorem.

Cf. A000045, A061899, A065108, A227875.

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

A065726 Primes p whose base-8 expansion is also the decimal expansion of a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 31, 43, 59, 67, 71, 89, 137, 151, 179, 191, 199, 223, 251, 257, 281, 283, 307, 311, 337, 353, 359, 367, 383, 409, 419, 433, 443, 449, 523, 563, 617, 619, 641, 659, 727, 787, 809, 811, 857, 887, 907, 919, 947, 977, 1033, 1039, 1097, 1123

Offset: 1

Patrick De Geest, Nov 15 2001

In general rebase notation (Marc LeBrun): p8 = (8) [p] (10).

			E.g., 787_10 = 1423_8 is prime, and so is 1423_10.

Harry J. Smith, 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

Primes in A036963.
Cf. A065720 up to A065727, A065361.
Cf. A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[ Range[2500], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 8]]] & ]

is(p, b=10, c=8)=isprime(vector(#d=digits(p, c), i, b^(#d-i))*d~)&&isprime(p) \\ This code can be used for other bases b, c when b>c. See A235265 for code also valid for bM. F. Hasler, Jan 12 2014

Definition clarified by M. F. Hasler, Jan 12 2014

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)

A235635 Primes whose base-5 representation is also the base-7 representation of a prime.

Original entry on oeis.org

2, 3, 5, 13, 17, 23, 29, 41, 43, 47, 53, 59, 61, 71, 79, 83, 101, 103, 137, 157, 163, 181, 191, 223, 227, 239, 281, 347, 379, 383, 419, 443, 463, 479, 547, 563, 571, 593, 641, 691, 701, 743, 757, 811, 839, 863, 877, 967, 997, 1049, 1051, 1087, 1097, 1109, 1151, 1171, 1217, 1249, 1259, 1283

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 = 32_5 and 32_7 = 23 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. A235627, 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], 7] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=7,c=5)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

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.

A235478 Primes whose base-2 representation also is the base-8 representation of a prime.

Original entry on oeis.org

7, 11, 13, 29, 37, 43, 47, 53, 61, 67, 71, 73, 107, 139, 149, 199, 211, 227, 263, 293, 307, 311, 317, 331, 347, 383, 389, 421, 461, 467, 541, 593, 601, 619, 641, 643, 739, 811, 863, 907, 937, 1061, 1069, 1093, 1117, 1163, 1223, 1283, 1301, 1319, 1321, 1409, 1433, 1439, 1489, 1499, 1523, 1559, 1619, 1697, 1811, 1861, 1879

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.
Appears to be a subsequence of A050150, A062090 and A216285.

			11 = 1011_2 and 1011_8 = 521 are both prime, so 11 is a term.

Robert Price, Table of n, a(n) for n = 1..2265
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235465 ⊂ A077722, A235266, A152079, A235475 - A235479, 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],8]]&] (* Harvey P. Dale, Sep 25 2015 *)

is(p,b=8)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

cp's OEIS Frontend

A235383 Fibonacci numbers that are the product of other Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

A065726 Primes p whose base-8 expansion is also the decimal expansion of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A235635 Primes whose base-5 representation is also the base-7 representation of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views