A065361 -id:A065361 - cp's OEIS frontend

A065724 Primes p such that the decimal expansion of its base-6 conversion is also prime.

2, 3, 5, 7, 19, 37, 67, 79, 97, 103, 127, 157, 163, 193, 229, 283, 307, 337, 439, 487, 547, 571, 601, 631, 643, 673, 733, 751, 757, 853, 877, 907, 937, 1021, 1033, 1039, 1087, 1093, 1117, 1171, 1249, 1279, 1423, 1567, 1627, 1663, 1723, 1753, 1831, 1873

Offset: 1

Patrick De Geest, Nov 15 2001

In general rebase notation (Marc LeBrun): p6 = (6) [p] (10).

			E.g., 1627_10 = 11311_6 is prime, and so is 11311_10.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Primes in A036959.

Cf. A065720 up to A065727, A065361.

Select[ Range[1900], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 6]]] & ]
Select[Prime[Range[300]],PrimeQ[FromDigits[IntegerDigits[#,6]]]&] (* Harvey P. Dale, Jul 17 2025 *)

isok(p) = isprime(p) && isprime(fromdigits(digits(p, 6), 10)); \\ Michel Marcus, Mar 05 2022

A065725 Primes p such that the decimal expansion of its base-7 conversion is also prime.

Original entry on oeis.org

2, 3, 5, 17, 29, 31, 43, 59, 71, 127, 157, 197, 211, 227, 239, 241, 337, 353, 367, 379, 409, 463, 491, 563, 577, 619, 647, 743, 757, 773, 787, 857, 911, 953, 967, 1093, 1123, 1163, 1193, 1249, 1303, 1373, 1429, 1459, 1471, 1499, 1583, 1597, 1613, 1627, 1669

Offset: 1

Patrick De Geest, Nov 15 2001

In general rebase notation (Marc LeBrun): p7 = (7) [p] (10).

			E.g., 787_10 = 2203_7 is prime, and so is 2203_10.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Primes in A036961.

Cf. A065720 up to A065727, A065361.

Select[ Range[2500], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 7]]] & ]
Select[Prime[Range[300]],PrimeQ[FromDigits[IntegerDigits[#,7]]]&] (* Harvey P. Dale, Nov 10 2022 *)

isok(p) = isprime(p) && isprime(fromdigits(digits(p, 7), 10)); \\ Michel Marcus, Mar 05 2022

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

A303787 a(n) = Sum_{i=0..m} d(i)4^i, where Sum_{i=0..m} d(i)5^i is the base-5 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 48, 49, 50, 51

Offset: 0

Seiichi Manyama, Apr 30 2018

			13 = 23_5, so a(13) = 2*4 + 3 = 11.
14 = 24_5, so a(14) = 2*4 + 4 = 12.
15 = 30_5, so a(15) = 3*4 + 0 = 12.
16 = 31_5, so a(16) = 3*4 + 1 = 13.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Sum_{i=0..m} d(i)*b^i, where Sum_{i=0..m} d(i)*(b+1)^i is the base (b+1) representation of n: A065361 (b=2), A215090 (b=3), this sequence (b=4), A303788 (b=5), A303789 (b=6).

Cf. A020654, A037459.

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 5)
        r += b * q
        b *= 4
    end
r end; [a(n) for n in 0:73] |> println # Peter Luschny, Jan 03 2021

a(n) = fromdigits(digits(n, 5), 4); \\ Michel Marcus, May 02 2018

def f(k, ary)
  (0..ary.size - 1).inject(0){|s, i| s + ary[i] * k ** i}
end
def A(k, n)
  (0..n).map{|i| f(k, i.to_s(k + 1).split('').map(&:to_i).reverse)}
end
p A(4, 100)

A303788 a(n) = Sum_{i=0..m} d(i)5^i, where Sum_{i=0..m} d(i)6^i is the base-6 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 30, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55

Offset: 0

Seiichi Manyama, Apr 30 2018

			16 = 24_6, so a(16) = 2*5 + 4 = 14.
17 = 25_6, so a(17) = 2*5 + 5 = 15.
18 = 30_6, so a(18) = 3*5 + 0 = 15.
19 = 31_6, so a(19) = 3*5 + 1 = 16.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Sum_{i=0..m} d(i)*b^i, where Sum_{i=0..m} d(i)*(b+1)^i is the base (b+1) representation of n: A065361 (b=2), A215090 (b=3), A303787 (b=4), this sequence (b=5), A303789 (b=6).

Cf. A037465.

a(n) = fromdigits(digits(n, 6), 5); \\ Michel Marcus, May 02 2018

def f(k, ary)
  (0..ary.size - 1).inject(0){|s, i| s + ary[i] * k ** i}
end
def A(k, n)
  (0..n).map{|i| f(k, i.to_s(k + 1).split('').map(&:to_i).reverse)}
end
p A(5, 100)

A303789 a(n) = Sum_{i=0..m} d(i)6^i, where Sum_{i=0..m} d(i)7^i is the base-7 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 53, 54, 54, 55

Offset: 0

Seiichi Manyama, Apr 30 2018

			19 = 25_7, so a(19) = 2*6 + 5 = 17.
20 = 26_7, so a(20) = 2*6 + 6 = 18.
21 = 30_7, so a(21) = 3*6 + 0 = 18.
22 = 31_7, so a(22) = 3*6 + 1 = 19.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Sum_{i=0..m} d(i)*b^i, where Sum_{i=0..m} d(i)*(b+1)^i is the base (b+1) representation of n: A065361 (b=2), A215090 (b=3), A303787 (b=4), A303788 (b=5), this sequence (b=6).

Cf. A037470.

a(n) = fromdigits(digits(n, 7), 6); \\ Michel Marcus, May 02 2018

def f(k, ary)
  (0..ary.size - 1).inject(0){|s, i| s + ary[i] * k ** i}
end
def A(k, n)
  (0..n).map{|i| f(k, i.to_s(k + 1).split('').map(&:to_i).reverse)}
end
p A(6, 100)

A065362 Rebase n from 4 to 2. Replace 4^k with 2^k in quaternary expansion of n.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17

Offset: 1

Marc LeBrun, Oct 31 2001

Notation: (4)[n](2).

			24 = 120 -> 1(4) + 2(2) + 0(1) = 8 = a(24).

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A065361.

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 4)
        r += b * q
        b *= 2
    end
r end; [a(n) for n in 0:79] |> println # Peter Luschny, Jan 03 2021

t = Table[FromDigits[RealDigits[n, 4], 2], {n, 1, 100}] (* Clark Kimberling, Aug 02 2012 *)

Rebase(x, b, c)= { local(d, e=0, f=1); while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=c); return(e) } { for (n=1, 1000, write("b065362.txt", n, " ", Rebase(n, 4, 2)) ) } \\ Harry J. Smith, Oct 17 2009

a(n) = 2*a(n/4) if n == 0 (mod 4); otherwise, a(n) = a(n-1) + 1. - Clark Kimberling, Aug 03 2012

A274515 a(n) is the number of times that the value of ternary n when read as hyperbinary occurs in the set of hyperbinary representations.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 2, 3, 3, 2, 3, 3, 1, 4, 4, 3, 5, 4, 3, 5, 5, 2, 5, 5, 3, 4, 4, 3, 5, 5, 2, 5, 5, 3, 4, 5, 3, 4, 4, 1, 5, 5, 4, 7, 5, 4, 7, 7, 3, 8, 8, 5, 7, 5, 4, 7, 7, 3, 8, 8, 5, 7, 8, 5, 7, 7, 2, 7, 7, 5, 8, 7, 5, 8, 8, 3, 7, 7, 4, 5, 5, 4, 7, 7, 3, 8

Offset: 0

Max Barrentine, Jun 25 2016

Stern's diatomic sequence A002487 counts the ways (n+1) can be represented if one allows 2's to be included in (n)'s binary representation (its "hyperbinary representations" in the terminology of A002487). A065361 maps ternary ordering onto these hyperbinary representations.

			5 in ternary is 12, which when read in hyperbinary is equal to 4. 6 in ternary is 20, which when read in hyperbinary is equal to 4. 9 in ternary is 100, which when read in hyperbinary is equal to 4. Since these are the only ways to represent 4 in hyperbinary, a(5) = a(6) = a(9) = 3.

Max Barrentine, Table of n, a(n) for n = 0..9841

Cf. A002487, A065361.

a(n) = A002487(A065361(n) + 1).

cp's OEIS Frontend

A065724 Primes p such that the decimal expansion of its base-6 conversion is also prime.

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A065725 Primes p such that the decimal expansion of its base-7 conversion is also prime.

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

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Extensions

A303787 a(n) = Sum_{i=0..m} d(i)*4^i, where Sum_{i=0..m} d(i)*5^i is the base-5 representation of n.

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A303788 a(n) = Sum_{i=0..m} d(i)*5^i, where Sum_{i=0..m} d(i)*6^i is the base-6 representation of n.

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Ruby

A303789 a(n) = Sum_{i=0..m} d(i)*6^i, where Sum_{i=0..m} d(i)*7^i is the base-7 representation of n.

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A065362 Rebase n from 4 to 2. Replace 4^k with 2^k in quaternary expansion of n.

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Formula

A274515 a(n) is the number of times that the value of ternary n when read as hyperbinary occurs in the set of hyperbinary representations.

A303787 a(n) = Sum_{i=0..m} d(i)4^i, where Sum_{i=0..m} d(i)5^i is the base-5 representation of n.

A303788 a(n) = Sum_{i=0..m} d(i)5^i, where Sum_{i=0..m} d(i)6^i is the base-6 representation of n.

A303789 a(n) = Sum_{i=0..m} d(i)6^i, where Sum_{i=0..m} d(i)7^i is the base-7 representation of n.