A098957 -id:A098957 - cp's OEIS frontend

A143245 Primes in A098957.

3, 5, 7, 13, 11, 17, 29, 23, 31, 41, 37, 53, 61, 43, 47, 97, 113, 73, 101, 67, 83, 107, 71, 127, 193, 233, 197, 229, 181, 173, 131, 163, 227, 251, 199, 167, 151, 223, 257, 449, 353, 337, 433, 409, 313, 421, 277, 373, 269, 461, 349, 509, 307, 331, 491, 283, 443

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 21 2008

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gray Code

Cf. A098957.

brev:= proc(n) local L;
  L:= convert(n,base,2);
  add(L[-i]*2^(i-1),i=1..nops(L))
end proc:
select(isprime, [seq(brev(ithprime(i)),i=1..100)]); # Robert Israel, Apr 03 2019

GrayCodeList[k_] := Module[{b = IntegerDigits[k, 2], i}, Do[ If[b[[i - 1]] == 1, b[[i]] = 1 - b[[i]]], {i, Length[b], 2, -1} ]; b ]; a[n_] = GrayCodeList[Prime[n]]; a0 = Table[Sum[a[n][[m + 1]]*2^m, {m, 0, Length[a[n]] - 1}], {n, 1, 200}]; Flatten[Table[If[PrimeQ[a0[[n]]], a0[[n]], {}], {n, 1, 200}]]

forprime(p=2, 1e3, v=binary(p); s=0; forstep(i=#v, 1, -1, s+=s+v[i]); if(isprime(s),print1(s", "))) \\ Charles R Greathouse IV, Nov 07 2011

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)).

A265326 n-th prime minus its binary reversal.

Original entry on oeis.org

1, 0, 0, 0, -2, 2, 0, -6, -6, 6, 0, -4, 4, -10, -14, 10, 4, 14, -30, -42, 0, -42, -18, 12, 30, 18, -12, 0, 18, 42, 0, -62, -8, -70, -20, -82, -28, -34, -62, -8, -26, 8, -62, 62, 34, -28, 8, -28, 28, 62, 82, -8, 98, 28, 0, -186, -84, -210, -60

Offset: 1

Max Barrentine, Dec 07 2015

a(n) = 0 iff A000040(n) is in A016041. - Altug Alkan, Dec 07 2015

The graph consists of a succession of parallelograms. The parallelograms end when there is a long run of mostly positive terms followed by a long run of mostly negative terms. The places where the successive parallelograms end are the primes just before a power of 2: 3, 7, 13, 31, 61, 127, 251, 509, 1021, 2039, 4093, 8191, 16381, 32749, ..., which are terms with indices 2, 4, 6, 11, 18, 31, 54, 97, 172, 309, 564, 1028, 1900, 3512, 6542, 12251, 23000, 43390, 82025, ... (see A014234 and A007053). - N. J. A. Sloane, May 29 2016

			n=5: prime(5) = 11_10 = 1011_2, reversing gives 1101_2 = 13_10, so a(5) = 11-13 = -2.

N. J. A. Sloane, Table of n, a(n) for n = 1..23000, May 29 2016 [First 10000 terms from _Robert Israel_]
Rémy Sigrist, Colored scatterplot of the first 82025 terms (corresponding to the prime numbers < 2^20) (where the color is function of A000040(n) mod 8)
N. J. A. Sloane, Table of n, a(n) for n = 1..78498
N. J. A. Sloane and Brady Haran, Amazing Graphs, Numberphile video (2019)

Cf. A055945, A068396, A098957, A007053, A014234.

revdigs:= proc(n) local L, j;
  L:= convert(n,base,2);
  add(L[-j]*2^(j-1),j=1..nops(L))
end proc:
map(t -> t - revdigs(t),  select(isprime, [2,seq(i,i=3..1000,2)])); # Robert Israel, Dec 08 2015

Table[# - FromDigits[Reverse@ IntegerDigits[#, 2], 2] &@ Prime@ n, {n, 60}] (* Michael De Vlieger, Dec 09 2015 *)

a098957(n) = my(v=binary(prime(n)), s); forstep(i=#v, 1, -1, s+=s+v[i]); s
a(n) = prime(n) - a098957(n); \\ Altug Alkan, Dec 07 2015

a(n) = A000040(n) - A098957(n).

a(n) = A055945(A000040(n)). - Michel Marcus, Dec 08 2015

A143257 Reverse binary expansion of the factorial numbers.

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 45, 441, 441, 3213, 16059, 172569, 517671, 6695325, 43746885, 903732249, 903732249, 14611840389, 110769926061, 1248788195355, 12439562858721, 154437141889677, 1902100636851663, 51339101124195573, 255363550208935617, 4458722921105347251

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 21 2008

Cf. A000142, A030101, A098957.

A143257 := proc(n)
    A030101(n!) ;
end proc:
seq(A143257(n),n=0..10) ; # R. J. Mathar, Mar 10 2015

a[n_] := FromDigits[ Reverse@ IntegerDigits[n!, 2], 2];
Array[a, 23] (* Robert G. Wilson v, Mar 11 2015 *)

a(n) = my(v=binary(n!), s); forstep(i=#v, 1, -1, s+=s+v[i]); s \\ Michel Marcus, May 18 2013

a(n) = A030101(n!).

Edited and renamed by Michel Marcus, May 18 2013

A159006 Transformation of prime(n): flip digits in the binary representation, revert the sequence of digits, and convert back to decimal.

Original entry on oeis.org

2, 0, 2, 0, 2, 4, 14, 6, 2, 8, 0, 22, 26, 10, 2, 20, 8, 16, 30, 14, 54, 6, 26, 50, 60, 44, 12, 20, 36, 56, 0, 62, 110, 46, 86, 22, 70, 58, 26, 74, 50, 82, 2, 124, 92, 28, 52, 4, 56, 88, 104, 8, 112, 32, 254, 62, 158, 30, 174, 206, 78, 182, 102, 38, 198, 134, 90, 234, 74, 138, 242

Offset: 1

Paolo P. Lava & Giorgio Balzarotti, Apr 02 2009

Write the n-th prime in binary as in A004676. Change all 0's to 1's and all 1's to 0's to reach A145382. Read the binary digits from right to left. a(n) is the decimal equivalent of the result.

All entries are even, i.e., members of A005843.

			37->100101->change digits 011010->read from right to left 010110->22 281->100011001->change digits 011100110->read from right to left 011001110->206

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A036044, A098957, A145382.

P:=proc(i) local a,b,j,k,n; for n from 1 by 1 to i do a:=convert(ithprime(n), binary); j:=length(a); b:=convert(a,string); k:=""; while j>0 do if substring(b,j)="1" then k:=cat(k,"0"); else k:=cat(k,"1"); fi; j:=j-1; od; a:=convert(k,decimal,binary); print(a); od; end: P(100);

FromDigits[Reverse@ BitNot@ IntegerDigits[#, 2] + 2, 2] & /@ Prime@ Range@ 71 (* Michael De Vlieger, Apr 22 2015 *)

a(n)=fromdigits(Vecrev(apply(n->1-n,binary(prime(n)))),2) \\ Charles R Greathouse IV, Apr 22 2015

from sympy import prime
def A159006(n): return -int((s:=bin(prime(n))[-1:1:-1]),2)-1+2**len(s) # Chai Wah Wu, Feb 04 2022

a(n) = A036044(prime(n)). - Michel Marcus, Apr 22 2015

Edited by R. J. Mathar, Apr 06 2009

A351528 Prime numbers ordered by their binary reversal.

Original entry on oeis.org

2, 3, 5, 7, 13, 11, 17, 29, 19, 23, 31, 41, 37, 53, 61, 43, 59, 47, 97, 113, 73, 89, 101, 109, 67, 83, 107, 71, 103, 79, 127, 193, 241, 137, 233, 197, 229, 149, 181, 173, 157, 131, 163, 227, 211, 179, 139, 251, 199, 167, 151, 239, 223, 191, 257, 449, 353, 401

Offset: 1

Rémy Sigrist, Feb 13 2022

See A104154 for the base-10 variant.

Mersenne primes (A000668) and Fermat primes (A019434) appear at their natural position.

			The first terms, alongside their binary and reverse binary expansions, are:
  n   a(n)  bin(a(n))  bin(rev(a(n)))
  --  ----  ---------  --------------
   1     2         10               1
   2     3         11              11
   3     5        101             101
   4     7        111             111
   5    13       1101            1011
   6    11       1011            1101
   7    17      10001           10001
   8    29      11101           10111
   9    19      10011           11001
  10    23      10111           11101
  11    31      11111           11111

Rémy Sigrist, Table of n, a(n) for n = 1..6542

Cf. A000668, A019434, A030101, A098957, A104154.

SortBy[Select[Range[2^9], PrimeQ], IntegerReverse[#, 2] &] (* Amiram Eldar, Feb 15 2022 *)

rev(n) = fromdigits(Vecrev(binary(n)), 2)
print (vecsort(primes([1, 2^9]), (p,q)->rev(p)-rev(q))[1..58])

from itertools import count, islice
from sympy import primerange
def A351528_gen(): # generator of terms
    yield from (int(d[::-1],2) for l in count(1) for d in sorted(bin(m)[:1:-1] for m in primerange(2**(l-1),2**l)))
A351528_list = list(islice(A351528_gen(),20)) # Chai Wah Wu, Feb 17 2022

A143250 Reverse binary expansion of the Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 3, 5, 1, 11, 21, 17, 59, 77, 9, 151, 317, 281, 879, 1507, 389, 5441, 5835, 4309, 31313, 18427, 1197, 69961, 71863, 68605, 423129, 354527, 83155, 1534501, 2633505, 1177835, 7573621, 9646897, 887751, 20771357, 38811817, 18203279, 110884035

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 21 2008

Old name was "Sequence of sum of Gray code Binary digits for Fibonacci sequence A000045: a(n) = GrayCodeBinarySum(A000045(n))".

Cf. A000045, A030101, A098957.

[Seqint(Reverse(Intseq(Fibonacci(n),2)),2): n in [1..40]]; // Bruno Berselli, May 01 2013

GrayCodeList[k_] := Module[{b = IntegerDigits[k, 2], i}, Do[ If[b[[i - 1]] == 1, b[[i]] = 1 - b[[i]]], {i, Length[b], 2, -1} ]; b ]; a[n_] = GrayCodeList[Fibonacci[n]]; a0 = Table[Sum[a[n][[m + 1]]*2^m, {m, 0, Length[a[n]] - 1}], {n, 1, 200}]
Table[FromDigits[Reverse[IntegerDigits[Fibonacci[n], 2]], 2], {n, 40}] (* Bruno Berselli, May 01 2013 *)

a(n) = {my(v=binary(fibonacci(n)), s); forstep(i=#v, 1, -1, s+=s+v[i]); return (s);} \\ Michel Marcus, May 01 2013

a(n) = Decimal(reverse(Binary(A000045(n)))).

Edited and new name by Michel Marcus, May 01 2013

cp's OEIS Frontend

A143245 Primes in A098957.

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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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A265326 n-th prime minus its binary reversal.

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A143257 Reverse binary expansion of the factorial numbers.

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A159006 Transformation of prime(n): flip digits in the binary representation, revert the sequence of digits, and convert back to decimal.

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A351528 Prime numbers ordered by their binary reversal.

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Mathematica

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A143250 Reverse binary expansion of the Fibonacci numbers.

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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.