A074832 -id:A074832 - cp's OEIS frontend

A235027 Reverse the bits of prime divisors of n (with 2 -> 2), and multiply together: a(0)=0, a(1)=1, a(2)=2, a(p) = revbits(p) for odd primes p, a(uv) = a(u) a(v) for composites.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 25, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 31, 32, 39, 34, 35, 36, 41, 50, 33, 40, 37, 42, 53, 52, 45, 58, 61, 48, 49, 50, 51, 44, 43, 54, 65, 56, 75, 46, 55, 60, 47, 62, 63, 64, 55, 78, 97

Offset: 0

Antti Karttunen, Jan 02 2014

This is not a permutation of integers: a(25) = 25 = 5*5 = a(19) is the first case which breaks the injectivity. However, the first 24 terms are equal with A057889, which is a GF(2)[X]-analog of this sequence and which in contrary to this, is bijective. This stems from the fact that the set of irreducible GF(2)[X] polynomials (A014580) is closed under bit-reversal (A056539), while primes (A000040) are not.
Sequence A290078 gives the positions n where the ratio a(n)/n obtains new record values.
Note, instead of A056539 we could as well use A057889 to reverse the bits of n, and also A030101 when restricted to odd primes.

			a(33) = a(3*11) = a(3) * a(11) = 3 * 13 = 39 (because 3, in binary '11', stays same when reversed, while 11 (eleven), in binary '1011', changes to '1101' = 13).

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n.
Index to divisibility sequences.

A235028 gives the fixed points. A235030 numbers such that n <> a(a(n)), or equally A001222(a(n)) > A001222(n). A235145 the number of iterations needed to reach a fixed point or cycle of 2, A235146 its records.

Cf. also A010051, A020639, A030101, A056539, A057889, A074832, A098957, A091214, A204219, A290078.

f[p_, e_] := IntegerReverse[p, 2]^e; f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Sep 03 2023 *)

revbits(n) = fromdigits(Vecrev(binary(n)), 2);
a(n) = {my(f = factor(n)); for (k=1, #f~, if (f[k,1] != 2, f[k,1] = revbits(f[k,1]););); factorback(f);} \\ Michel Marcus, Aug 05 2017

Completely multiplicative with a(0)=0, a(1)=1, a(p) = A056539(p) for primes p (which maps 2 to 2, and reverses the binary representation of odd primes), and a(u*v) = a(u) * a(v) for composites.
Equally, after a(0)=0, a(p * q * ... * r) = A056539(p) * A056539(q) * ... * A056539(r), for primes p, q, etc., not necessarily distinct.
a(0)=0, a(1)=1, a(n) = A056539(A020639(n)) * a(n/A020639(n)).

A235145 a(n) = Number of steps to reach a fixed point or 2-cycle, when iterating A235027 starting from value n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2

Offset: 0

Antti Karttunen, Jan 03 2014

Equally, a(n) = minimum number of steps needed to repeat k = A235027(k)(starting from k = n) until A001222(A235027(k)) = A001222(k).

Or in other words, how many times are needed to repeatedly factorize the number, to reverse the bits of each odd prime factor (with A056539) and factorize and bit-reverse the reversed factors again, until the number of prime divisors no more grows, meaning that we have found either a fixed point or entered a cycle of two.

			19, '10011' in binary, when reversed, yields '11001' = 25, when factored, yields 5 * 5, ('101' * '101' in binary), which divisors stay same when reversed, thus it took one iteration step to reach a point where the number of prime divisors no more grows. Thus a(19)=1.

Antti Karttunen, Table of n, a(n) for n = 0..10001

A235146 gives the positions of records. Cf. A001222, A056539, A074832, A235027.

revbits(n) = fromdigits(Vecrev(binary(n)), 2);
a235027(n) = {f = factor(n); for (k=1, #f~, if (f[k,1] != 2, f[k,1] = revbits(f[k,1]););); factorback(f);}
find(v, newn) = {for (k=1, #v, if (v[#v -k + 1] == newn, return (k));); return (0);}
a(n) = {ok = 0; v = [n]; while (! ok, newn = a235027(n); ind = find(v, newn); if (ind, ok = 1, v = concat(v, newn); n = newn);); #v - ind;} \\ Michel Marcus, Aug 06 2017

If A235027(A235027(n)) = n, a(n)=0, otherwise 1+a(A235027(n)).
Equally, if A001222(A235027(n)) = A001222(n), a(n)=0, otherwise 1+a(A235027(n)).
a(2n) = a(n), and in general, for composite values a(u * v) = max(a(u),a(v)).
For composite n, a(n) = Max_{p|n} a(p). [The above reduces to this: select the maximal value from all values a(p) computed for primes p dividing n]
For prime p, a(p) = 0 if A056539(p) is also prime (p is 2 or in A074832), otherwise a(p) = 1+a(A056539(p)).

A074833 Primes whose ternary reversal is also prime.

Original entry on oeis.org

2, 5, 7, 11, 13, 19, 23, 31, 37, 41, 47, 53, 61, 67, 79, 83, 101, 103, 113, 127, 131, 151, 163, 167, 173, 179, 181, 191, 193, 211, 227, 233, 263, 281, 293, 311, 331, 349, 353, 359, 401, 409, 419, 421, 431, 439, 449, 463, 467, 491, 499, 503, 521, 523, 541, 563

Offset: 1

Robert G. Wilson v, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007500, A074832, A074834, A075235, A075236, A075237, A075238, A075239.

Prime[ Select[ Range[110], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ Prime[ # ], 3]], 3]] &]]

A074834 Primes whose base 4 reversal is also prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 29, 31, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 107, 193, 197, 199, 211, 233, 239, 241, 251, 257, 269, 277, 281, 293, 311, 313, 337, 353, 367, 373, 383, 397, 401, 409, 419, 433, 443, 449, 457, 461, 467, 487, 491, 499, 509, 787, 797, 809

Offset: 1

Robert G. Wilson v, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007500, A074832, A074833, A075235, A075236, A075237, A075238, A075239.

Prime[ Select[ Range[140], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ Prime[ # ], 4]], 4]] &]]

A075235 Primes whose base 5 reversal is also prime.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 149, 151, 157, 163, 167, 191, 193, 211, 223, 227, 229, 233, 239, 251, 257, 269, 271, 277, 281, 293, 317, 331, 337, 347, 349, 353, 359, 367

Offset: 1

Robert G. Wilson v, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007500, A074832, A074833, A074834, A075236, A075237, A075238, A075239.

Prime[ Select[ Range[100], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ Prime[ # ], 5]], 5]] &]]

A075236 Primes whose base 6 reversal is also prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 37, 41, 43, 53, 59, 61, 67, 71, 181, 191, 193, 197, 199, 211, 227, 233, 257, 263, 271, 277, 281, 293, 307, 311, 313, 317, 331, 337, 349, 359, 367, 373, 379, 383, 389, 431, 1087, 1093, 1103, 1109, 1117, 1123, 1153, 1187, 1193, 1201, 1213

Offset: 1

Robert G. Wilson v, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007500, A074832, A074833, A074834, A075235, A075237, A075238, A075239.

Prime[ Select[ Range[200], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ Prime[ # ], 6]], 6]] &]]
Select[Prime[Range[200]],PrimeQ[FromDigits[Reverse[IntegerDigits[#,6]],6]]&] (* Harvey P. Dale, Oct 12 2023 *)

A075237 Primes whose base 7 reversal is also prime.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 67, 71, 79, 89, 97, 101, 107, 127, 137, 139, 149, 151, 157, 167, 173, 179, 193, 197, 199, 211, 227, 229, 233, 241, 257, 269, 271, 277, 281, 307, 311, 331, 337, 347, 373, 389, 397, 401, 419, 421, 433, 439, 443

Offset: 1

Robert G. Wilson v, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007500, A074832, A074833, A074834, A075235, A075236, A075238, A075239.

Prime[ Select[ Range[100], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ Prime[ # ], 7]], 7]] &]]

A075238 Primes whose base 8 reversal is also prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 31, 41, 43, 47, 59, 61, 67, 71, 73, 79, 89, 97, 101, 107, 113, 193, 211, 227, 233, 239, 251, 349, 353, 373, 383, 449, 457, 463, 479, 487, 491, 503, 509, 521, 523, 541, 577, 587, 643, 677, 683, 719, 733, 751, 757, 773, 787, 811, 823, 827, 829

Offset: 1

Robert G. Wilson v, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A007500, A074832, A074833, A074834, A075235, A075236, A075237, A075239.

Prime[ Select[ Range[150], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ Prime[ # ], 8]], 8]] &]]
Select[Prime[Range[200]],PrimeQ[IntegerReverse[#,8]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 09 2017 *)

A075239 Primes whose base 9 reversal is also prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47, 67, 71, 73, 79, 83, 97, 101, 109, 113, 127, 139, 151, 157, 163, 173, 179, 181, 191, 193, 197, 227, 229, 239, 241, 331, 337, 353, 367, 373, 379, 383, 389, 397, 419, 433, 439, 457, 463, 479, 571, 577, 593, 599, 601, 607

Offset: 1

Robert G. Wilson v, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007500, A074832, A074833, A074834, A075235, A075236, A075237, A075238.

Prime[ Select[ Range[115], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ Prime[ # ], 9]], 9]] &]]

A080790 Binary emirps, primes whose binary reversal is a different prime.

Original entry on oeis.org

11, 13, 23, 29, 37, 41, 43, 47, 53, 61, 67, 71, 83, 97, 101, 113, 131, 151, 163, 167, 173, 181, 193, 197, 199, 223, 227, 229, 233, 251, 263, 269, 277, 283, 307, 331, 337, 349, 353, 359, 373, 383, 409, 421, 431, 433, 449, 461, 463, 479, 487, 491, 503, 509, 521

Offset: 1

Reinhard Zumkeller, Mar 25 2003

Members of A074832 that are not in A006995. - Robert Israel, Aug 31 2016

			A000040(10) = 29 -> '11101' rev '10111' -> 23 = A000040(9), therefore 29 and 23 are terms.
The prime 19 is not a term, as 19 -> '10011' rev '11001' -> 25 = 5^2; and 7 = A074832(3) is not a term because it is a binary palindrome (A006995) and therefore not different.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A006567, A006995, A074832, A004676, A007088.

revdigs:= proc(n) local L; L:= convert(n,base,2); add(L[-i]*2^(i-1),i=1..nops(L)) end proc:
filter:= proc(t) local r; if not isprime(t) then return false fi;
  r:= revdigs(t); r <> t and isprime(r) end proc:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Aug 30 2016

Select[Prime[Range[100]], (r = IntegerReverse[#, 2]) != # && PrimeQ[r] &] (* Amiram Eldar, Jul 28 2025 *)

from sympy import isprime
def ok(n):
    r = int(bin(n)[2:][::-1], 2)
    return n != r and isprime(n) and isprime(r)
print([k for k in range(600) if ok(k)]) # Michael S. Branicky, Jul 30 2022

cp's OEIS Frontend

A235027 Reverse the bits of prime divisors of n (with 2 -> 2), and multiply together: a(0)=0, a(1)=1, a(2)=2, a(p) = revbits(p) for odd primes p, a(u*v) = a(u) * a(v) for composites.

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A235145 a(n) = Number of steps to reach a fixed point or 2-cycle, when iterating A235027 starting from value n.

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A074833 Primes whose ternary reversal is also prime.

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A074834 Primes whose base 4 reversal is also prime.

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A075235 Primes whose base 5 reversal is also prime.

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A075236 Primes whose base 6 reversal is also prime.

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A075237 Primes whose base 7 reversal is also prime.

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A075238 Primes whose base 8 reversal is also prime.

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A075239 Primes whose base 9 reversal is also prime.

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A235027 Reverse the bits of prime divisors of n (with 2 -> 2), and multiply together: a(0)=0, a(1)=1, a(2)=2, a(p) = revbits(p) for odd primes p, a(uv) = a(u) a(v) for composites.