A000523 -id:A000523 - cp's OEIS frontend

A098390 Prime(n)+Log2(prime(n)), where Log2=A000523.

3, 4, 7, 9, 14, 16, 21, 23, 27, 33, 35, 42, 46, 48, 52, 58, 64, 66, 73, 77, 79, 85, 89, 95, 103, 107, 109, 113, 115, 119, 133, 138, 144, 146, 156, 158, 164, 170, 174, 180, 186, 188, 198, 200, 204, 206, 218, 230, 234, 236, 240, 246, 248, 258, 265, 271, 277, 279

Offset: 1

Reinhard Zumkeller, Sep 06 2004

nonn

a(n) = A000040(n) + A098388(n).

			a(10) = A000040(10) + A098388(10) = 29 + 4 = 33.

Cf. A098389, A014688, A098387, A098393.

#+Floor[Log[2,#]]&/@Prime[Range[60]] (* Harvey P. Dale, Dec 30 2011 *)

A080316 a(n) = A080315(n) - 2^A000523(A080315(n)), i.e., the terms of A080315 without their most significant bit.

Original entry on oeis.org

0, 12, 204, 240, 3276, 3312, 4032, 3852, 3888, 52428, 52464, 53184, 53004, 53040, 64524, 64560, 64704, 61644, 61680, 65280, 62400, 62220, 62256, 838860, 838896, 839616, 839436, 839472, 850956, 850992, 851136, 848076, 848112, 851712, 848832

Offset: 0

Antti Karttunen, Mar 02 2003

nonn

Same sequence in binary: A080317.

A096116 a(1)=1, if n=(2^k)+1, a(n) = k+2, otherwise a(n) = 2+A000523(n-1)+a(2+A035327(n-1)).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 7, 6, 5, 11, 12, 14, 9, 10, 8, 7, 6, 13, 14, 16, 15, 20, 18, 17, 11, 12, 13, 15, 10, 11, 9, 8, 7, 15, 16, 18, 17, 22, 20, 19, 18, 24, 25, 27, 22, 23, 21, 20, 13, 14, 15, 17, 16, 21, 19, 18, 12, 13, 14, 16, 11, 12, 10, 9, 8, 17, 18, 20, 19, 24, 22, 21, 20, 26

Offset: 1

Amarnath Murthy, Jun 30 2004

nonn

Each n > 1 occurs A025147(n) times in the sequence.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A096111, A050029, A050030, A052330, A096113, A096114, A096115.

a = {1}; Do[AppendTo[a, If[BitAnd[n - 1, n - 2] == 0, Log2[n - 1] + 2, 2 + Floor[Log2[n - 1]] + a[[2 + BitXor[n - 1, 2^Ceiling[Log2[n]] - 1]]]]], {n, 2, 74}]; a (* Ivan Neretin, Jun 24 2016 *)

(define (A096116 n) (cond ((= 1 n) 1) ((pow2? (- n 1)) (+ 2 (A000523 (- n 1)))) (else (+ 2 (A000523 (- n 1)) (A096116 (+ 2 (A035327 (- n 1))))))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))
;; Antti Karttunen, Aug 25 2006

Edited and extended by Antti Karttunen, Aug 25 2006

A098386 a(n) = prime(n)-Log2(n), where Log2 = A000523.

Original entry on oeis.org

2, 2, 4, 5, 9, 11, 15, 16, 20, 26, 28, 34, 38, 40, 44, 49, 55, 57, 63, 67, 69, 75, 79, 85, 93, 97, 99, 103, 105, 109, 123, 126, 132, 134, 144, 146, 152, 158, 162, 168, 174, 176, 186, 188, 192, 194, 206, 218, 222, 224, 228, 234, 236, 246, 252, 258, 264, 266, 272, 276

Offset: 1

Reinhard Zumkeller, Sep 06 2004

nonn

			a(10) = A000040(10) - A000523(10) = 29 - 3 = 26.

Cf. A000040, A000523, A098387, A014689, A098389, A098392.

Table[Prime[n]-Floor[Log2[n]],{n,60}] (* Harvey P. Dale, Nov 20 2021 *)

A098387 Prime(n)+Log2(n), where Log2=A000523.

Original entry on oeis.org

2, 4, 6, 9, 13, 15, 19, 22, 26, 32, 34, 40, 44, 46, 50, 57, 63, 65, 71, 75, 77, 83, 87, 93, 101, 105, 107, 111, 113, 117, 131, 136, 142, 144, 154, 156, 162, 168, 172, 178, 184, 186, 196, 198, 202, 204, 216, 228, 232, 234, 238, 244, 246, 256, 262, 268, 274, 276

Offset: 1

Reinhard Zumkeller, Sep 06 2004

nonn

			a(10) = A000040(10) + A000523(10) = 29 + 3 = 32.

Cf. A098386, A014688, A098390, A098393.

A268726 Index of the toggled bit between n and A268717(n+1): a(n) = A000523(A003987(n, A268717(1+n))).

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 0, 1, 4, 0, 0, 1, 0, 1, 2, 0, 5, 0, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 3, 0, 0, 1, 6, 0, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 3, 0, 0, 1, 0, 1, 2, 0, 3, 0, 0, 1, 4, 0, 0, 1, 0, 1, 2, 0, 7, 0, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 3, 0, 0, 1, 0, 1, 2, 0, 3, 0, 0, 1, 4, 0, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 3, 0, 0, 1, 4, 0, 0, 1, 0, 1, 2, 0, 5, 0, 0, 1, 0, 1, 2, 0, 0

Offset: 0

Antti Karttunen, Feb 13 2016

A fractal sequence, because a permutation of A007814. Removing zeros yields A268727(n) = a(n)+1.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Indranil Ghosh, C program to generate the sequence

One less than A268727.
Cf. A003188, A006068.
Cf. A000523, A003987, A007814, A101080, A268717.
Cf. also array A268833.

A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m=A006068[Floor[n/2]]}, 2m + Mod[Mod[n,2] + Mod[m, 2], 2]]]; A268717[n_]:=If[n<1, 0, A003188[1 + A006068[n - 1]]]; a[n_]:= Floor[Log[2, BitXor[n, A268717[n + 1]]]]; Table[a[n], {n, 0, 200}] (* Indranil Ghosh, Apr 02 2017 *)

A003188(n) = bitxor(n, n\2);
A006068(n) = if(n<2, n, {my(m = A006068(n\2)); 2*m + (n%2 + m%2)%2});
A268717(n) = if(n<1, 0, A003188(1 + A006068(n - 1)));
for(n=0, 200, print1(logint(bitxor(n, A268717(n + 1)), 2),", "))  \\ Indranil Ghosh, Apr 02 2017

def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    else:
        m=A006068(n//2)
        return 2*m + (n%2 + m%2)%2
def A268717(n): return 0 if n<1 else A003188(1 + A006068(n - 1))
print([int(math.floor(math.log(n^A268717(n + 1), 2))) for n in range(201)]) # Indranil Ghosh, Apr 02 2017

(define (A268726 n) (A000523 (A003987bi n (A268717 (+ 1 n))))) ;; A003987bi implements the bitwise-XOR, defined in A003987.

a(n) = A007814(1 + A006068(n)).
a(n) = A000523(A003987(n, A268717(1+n))).
a(n) = floor(log_2(n XOR A003188(1 + A006068(n)))).
Other identities:
For all n >= 1, a(A003188(n-1)) = A007814(n).

A286574 Sum of the binary weights of the lengths of 1-runs in base-2 representation of n: a(n) = A000523(A286575(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 3, 2, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 3, 4, 2, 3, 3, 3, 3, 4, 2, 3, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 3, 4, 1

Offset: 0

Antti Karttunen, May 28 2017

a(0) = 0 (an empty sum).

			For n = 27, "11011" in binary, there are two 1-runs, both of length 2, thus a(27) = A000120(2) + A000120(2) = 1 + 1 = 2.
For n = 29, "11101" in binary, there are two 1-runs, of lengths 1 and 3, thus a(29) = A000120(1) + A000120(3) = 1 + 2 = 3.
For n = 61, "111101" in binary, there are two 1-runs, of lengths 1 and 4, thus a(61) = A000120(1) + A000120(4) = 1 + 1 = 2.

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A000120, A000523, A001316, A005940, A064547, A286575.

from sympy import factorint, prime, log
import math
def wt(n): return bin(n).count("1")
def a037445(n):
    f=factorint(n)
    return 2**sum([wt(f[i]) for i in f])
def A(n): return n - 2**int(math.floor(log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a286575(n): return a037445(b(n))
def a(n): return int(math.floor(log(a286575(n), 2))) # Indranil Ghosh, May 30 2017

# uses RLT function from A278159
def A286574(n): return len(bin(RLT(n,lambda m: 2**(bin(m).count('1')))))-3 # Chai Wah Wu, Feb 04 2022

(define (A286574 n) (A000523 (A286575 n)))

a(n) = A000523(A286575(n)). [Log_2 of run-length transform of A001316.]

a(n) = A064547(A005940(1+n)).

A317359 a(0) = 0, a(1) = 1; for n >= 2, a(n) = freq(a(n-g(n)),n) where g = A000523 and freq(i, j) is the number of times i appears in the terms a(0) .. a(j-1).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3, 3, 8, 8, 8, 8, 4, 4, 4, 4, 7, 7, 7, 7, 4, 4, 4, 4, 4, 12, 12, 12, 12, 12, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 11, 11, 11, 11, 11, 5, 5, 5, 5, 5, 5, 17, 17, 17, 17, 17, 17, 6, 6, 6, 6, 6, 6, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 11, 11, 11, 11, 11

Offset: 0

Altug Alkan, Jul 26 2018

Alois P. Heinz, Table of n, a(n) for n = 0..65536
Altug Alkan, A line graph of a(n) for n <= 2500

Cf. A000523, A317223, A317361.

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= `if`(n<2, n, b(a(n-ilog2(n))));
      b(t):= b(t)+1; t
    end:
seq(a(n), n=0..200);  # Alois P. Heinz, Jul 26 2018

c = <||>; f[n_] := If[KeyExistsQ[c, n], c[n], 0]; a[n_] := a[n] = Block[{v}, v = If[n < 2, n, f[a[n - Floor@ Log2@ n]]]; If[f[v] > 0, c[v] = c[v] + 1, c[v] = 1]; v]; Array[a, 96, 0] (* Giovanni Resta, Jul 26 2018 *)

A324728 Binary length of A324712: a(n) = 0 if A324712(n) = 0, otherwise a(n) = 1+A000523(A324712(n)).

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 4, 4, 4, 4, 5, 4, 6, 5, 5, 5, 7, 3, 8, 5, 6, 6, 9, 5, 5, 7, 5, 6, 10, 6, 11, 6, 6, 8, 6, 6, 12, 9, 8, 6, 13, 5, 14, 7, 6, 10, 15, 6, 6, 4, 8, 8, 16, 3, 7, 7, 10, 11, 17, 5, 18, 12, 5, 7, 7, 7, 19, 9, 10, 5, 20, 7, 21, 13, 6, 10, 7, 7, 22, 7, 7, 14, 23, 7, 9, 15, 12, 8, 24, 7, 8, 11, 12, 16, 10, 7, 25, 5, 8, 7, 26, 9, 27, 9, 7

Offset: 1

Antti Karttunen, Mar 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000523, A323243, A324712, A324733, A324735, A324811.

A324712(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A323243(d)))); (v); }; \\ Needs also code from A323243.
A000523(n) = if( n<1, 0, #binary(n) - 1); \\ From A000523
A324728(n) = { my(k=A324712(n)); if(!k,k,(1+A000523(k))); };

If A324712(n) = 0, then a(n) = 0, otherwise a(n) = 1+A000523(A324712(n)).

a(A000040(n)) = n.

A324733 a(n) = 0 if A324712(n) = 0, otherwise a(n) = 1 + A000523(A324712(n)) - A007814(A324712(n)).

Original entry on oeis.org

0, 1, 2, 3, 3, 1, 4, 2, 4, 4, 5, 4, 6, 3, 4, 1, 7, 3, 8, 3, 6, 5, 9, 2, 5, 6, 3, 6, 10, 4, 11, 3, 5, 7, 5, 4, 12, 7, 6, 1, 13, 2, 14, 7, 6, 9, 15, 3, 6, 4, 1, 8, 16, 1, 7, 5, 8, 8, 17, 4, 18, 11, 3, 4, 6, 7, 19, 9, 9, 4, 20, 5, 21, 12, 6, 10, 6, 7, 22, 4, 3, 13, 23, 7, 6, 14, 10, 5, 24, 7, 8, 11, 11, 15, 8, 3, 25, 2, 8, 6, 26, 9, 27, 2, 5

Offset: 1

Antti Karttunen, Mar 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A323243, A324712, A324724, A324728, A324734, A324735.

A324712(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A323243(d)))); (v); }; \\ Needs also code from A323243.
A000523(n) = if( n<1, 0, #binary(n) - 1); \\ From A000523
A007814(n) = valuation(n,2)
A324733(n) = { my(k=A324712(n)); if(!k,k,1 + (A000523(k)-A007814(k))); };

If A324712(n) = 0, then a(n) = 0, otherwise a(n) = 1 + A324728(n) - A324724(n).

cp's OEIS Frontend

A098390 Prime(n)+Log2(prime(n)), where Log2=A000523.

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A080316 a(n) = A080315(n) - 2^A000523(A080315(n)), i.e., the terms of A080315 without their most significant bit.

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A096116 a(1)=1, if n=(2^k)+1, a(n) = k+2, otherwise a(n) = 2+A000523(n-1)+a(2+A035327(n-1)).

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A098386 a(n) = prime(n)-Log2(n), where Log2 = A000523.

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A098387 Prime(n)+Log2(n), where Log2=A000523.

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A268726 Index of the toggled bit between n and A268717(n+1): a(n) = A000523(A003987(n, A268717(1+n))).

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A286574 Sum of the binary weights of the lengths of 1-runs in base-2 representation of n: a(n) = A000523(A286575(n)).

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A317359 a(0) = 0, a(1) = 1; for n >= 2, a(n) = freq(a(n-g(n)),n) where g = A000523 and freq(i, j) is the number of times i appears in the terms a(0) .. a(j-1).

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A324728 Binary length of A324712: a(n) = 0 if A324712(n) = 0, otherwise a(n) = 1+A000523(A324712(n)).

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