A235146 -id:A235146 - 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)).

A235030 Numbers such that A235027(A235027(n)) <> n; Numbers which are divisible by any of the odd terms of A204219.

Original entry on oeis.org

19, 38, 57, 59, 76, 79, 89, 95, 103, 109, 114, 118, 133, 137, 139, 149, 152, 157, 158, 171, 177, 178, 179, 190, 191, 206, 209, 211, 218, 228, 236, 237, 239, 241, 247, 266, 267, 271, 274, 278, 281, 285, 293, 295, 298, 304, 309, 311, 314, 316, 317, 323, 327, 342

Offset: 1

Antti Karttunen, Jan 02 2014

Sequence consists of all the primes in A204219 (after 2), together with all of their multiples.
Note that this is not the same as the numbers that do not occur in A235027 ("Garden of Eden" numbers for A235027), a subsequence of this sequence, which begins as: 19, 38, 57, 59, 76, 79, 89, 95, 103, 109, 114, 118, 133, 137, 139, 149, 152, 157, 158, 171, 177, 178, 179, 190, 191, 206, 211, 218, 228, 236, 237, 239, 241, 266, 267, 271, 274, 278, 281, 285, 293, 298, 304, 309, 311, 314, 316, 317, 327, 342, 347, 354, 356, 358, ...
The first term that occurs in this sequence, but not in the "GoE"-sequence is a(27)=209, as a(139) = 209 = 11*19 and 139 = A235146(2), the least integer which requires two steps to reach a fixed point or 2-cycle.
Both the above "GoE"-sequence, and its differences from this will be submitted later.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A204219, A235027, A235145, A235146, A056539.

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.

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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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A235030 Numbers such that A235027(A235027(n)) <> n; Numbers which are divisible by any of the odd terms of A204219.

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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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A235030 Numbers such that A235027(A235027(n)) <> n; Numbers which are divisible by any of the odd terms of A204219.

Views

Author

Keywords

Comments

Links

Crossrefs

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.