A204219 -id:A204219 - cp's OEIS frontend

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

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.

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.

Original entry on oeis.org

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)).

A235146 a(n) = Least integer k such that it takes n iterations of "factor and reverse bits of odd prime divisors" (A235027) before a fixed point or cycle of 2 is reached; records in A235145.

Original entry on oeis.org

0, 19, 139, 719, 4793, 23773, 260863, 2375231, 21793843

Offset: 0

Antti Karttunen, Jan 03 2014

Note, as for all composite values A235145(u * v) = max(A235145(u), A235145(v)) which can be further reduced as A235145(n) = Max_{p|n} A235145(p), and because for any odd prime p, lpf(A056539(p)) >= 3 (where lpf = A020639, the least prime dividing n) while 1/2 < A056539(n)/n < 2, it follows that this sequence gives also the positions of the records in A235145, as its new values must appear in order.
Also, because of that multiplicativity criterion, all terms (after zero) must be primes, and specifically, the terms are a subset of A235030 (i.e., of A204219).
Conjecture: additional property is that the primes here belong to that subset of p in A204219 for which A056539(p) > p. The list of such primes begins as: 19, 79, 103, 137, 139, 149, 157, 179, 191, 239, 271, 281, 293, 311, 317, 347, 367, 379, 439, 523, 541, 547, 557, 563, 569, 587, 607, 613, 647, 659, 719, 733, 743, 751, 787, ...

A subset of A235030 and A204219.

Cf. A235145, A235027, A056539.

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);}
a235145(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;}
a(n) = {k = 0; while (a235145(k) != n, k = nextprime(k+1)); k;}
lista(nn) = {kprec = 0; for (n=0, nn, k = kprec; while (a235145(k) != n, k = nextprime(k+1)); print1(k, ", "); kprec = k;);} \\ Michel Marcus, Aug 06 2017

a(5)-a(8) from Michel Marcus, Aug 06 2017

A226019 Primes whose binary reversal is a square.

Original entry on oeis.org

2, 19, 79, 149, 569, 587, 1237, 2129, 2153, 2237, 2459, 2549, 4129, 4591, 4657, 4999, 8369, 8999, 9587, 9629, 9857, 10061, 17401, 17659, 17737, 18691, 20149, 20479, 33161, 33347, 34631, 35117, 35447, 39023, 40427, 40709, 66403, 68539, 74707, 75703, 79063, 79333, 80071

Offset: 1

Alex Ratushnyak, May 23 2013

The sequence of corresponding squares begins: 1, 25, 121, 169, 625, 841, 1369, 2209, 2401, 3025, 3481, 2809, 4225, 7921, ...

For n>1 the second and third most significant bits of a(n) are "0" because all odd squares are equal to 1 mod 8. - Andres Cicuttin, May 12 2016

Chai Wah Wu, Table of n, a(n) for n = 1..6182

Cf. A007488, A074832.

Subsequence of A204219. Cf. also A235027.

Select[Table[Prime[j],{j,1,10000}],Element[Sqrt[FromDigits[Reverse[IntegerDigits[#,2]],2]],Integers]&] (* Andres Cicuttin, May 12 2016 *)

isok(k) = isprime(k) && issquare(fromdigits(Vecrev(binary(k)), 2)); \\ Michel Marcus, Feb 19 2021

import math
primes = []
def addPrime(k):
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
  r = 0
  p = k
  while k:
    r = r*2 + (k&1)
    k>>=1
  s = int(math.sqrt(r))
  if s*s == r:  print(p, end=', ')
addPrime(2)
addPrime(3)
for i in range(5, 1000000000, 6):
  addPrime(i)
  addPrime(i+2)

from sympy import isprime
A226019_list, i, j = [2], 0, 0
while j < 2**34:
    p = int(format(j,'b')[::-1],2)
    if j % 2 and isprime(p):
        A226019_list.append(p)
    j += 2*i+1
    i += 1
A226019_list = sorted(A226019_list) # Chai Wah Wu, Dec 20 2015

from sympy import integer_nthroot, primerange
def ok(p): return integer_nthroot(int(bin(p)[:1:-1], 2), 2)[1]
def aupto(lim): return [p for p in primerange(2, lim+1) if ok(p)]
print(aupto(80071)) # Michael S. Branicky, Feb 19 2021

cp's OEIS Frontend

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

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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(u*v) = a(u) * a(v) for composites.

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A235146 a(n) = Least integer k such that it takes n iterations of "factor and reverse bits of odd prime divisors" (A235027) before a fixed point or cycle of 2 is reached; records in A235145.

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A226019 Primes whose binary reversal is a square.

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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.