A285097 -id:A285097 - cp's OEIS frontend

A285099 a(n) is the zero-based index of the second least significant 1-bit in the base-2 representation of n, or 0 if there are fewer than two 1-bits in n.

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

Offset: 0

Antti Karttunen, Apr 19 2017

			For n = 3, "11" in binary, the second least significant 1-bit (the second 1-bit from the right) is at position 1 (when the rightmost position is position 0), thus a(3) = 1.
For n = 4, "100" in binary, there is just one 1-bit present, thus a(4) = 0.
For n = 5, "101" in binary, the second 1-bit from the right is at position 2, thus a(5) = 2.
For n = 25, "11001" in binary, the second 1-bit from the right is at position 3, thus a(25) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A000120, A004198, A007814, A129760, A285097.

a007814[n_]:=IntegerExponent[n, 2]; a[n_]:=If[DigitCount[n, 2, 1]<2, 0, a007814[BitAnd[n, n - 1]]]; Table[a[n], {n, 0, 150}] (* Indranil Ghosh, Apr 20 2017 *)

import math
def a007814(n): return int(math.log(n - (n & n - 1), 2))
def a(n): return 0 if bin(n)[2:].count("1") < 2 else a007814(n & (n - 1)) # Indranil Ghosh, Apr 20 2017

(define (A285099 n) (if (<= (A000120 n) 1) 0 (A007814 (A004198bi n (- n 1))))) ;; A004198bi implements bitwise-and.

If A000120(n) < 2, a(n) = 0, otherwise a(n) = A007814(A129760(n)) = A007814(n AND (n-1)). [Where AND is bitwise-and, A004198].
From Jeffrey Shallit, Apr 19 2020: (Start)
This is a 2-regular sequence, satisfying the identities
a(4n) = -a(n) + a(2n),
a(4n+2) = a(4n+1),
a(8n+1) = -a(2n+1) + 2a(4n+1),
a(8n+3) = a(4n+3),
a(8n+5) = 2a(4n+3),
a(8n+7) = a(4n+3). (End)

A366370 Square array A(n,k) giving the length of the least significant run of 0-bits in binary expansion of A000225(n)^k, or 0 if A000225(n)^k is a binary repunit.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 1, 3, 0, 0, 3, 1, 4, 0, 0, 2, 4, 1, 5, 0, 0, 2, 2, 5, 1, 6, 0, 0, 1, 3, 2, 6, 1, 7, 0, 0, 4, 1, 4, 2, 7, 1, 8, 0, 0, 3, 5, 1, 5, 2, 8, 1, 9, 0, 0, 2, 3, 6, 1, 6, 2, 9, 1, 10, 0, 0, 1, 3, 3, 7, 1, 7, 2, 10, 1, 11, 0, 0, 3, 1, 4, 3, 8, 1, 8, 2, 11, 1, 12, 0, 0, 2, 4, 1, 5, 3, 9, 1, 9, 2, 12, 1, 13, 0

Offset: 1

Antti Karttunen, Oct 14 2023

			The top left corner of the square array:
  n\k| 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17
-----+-------------------------------------------------------------------
   1 | 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,
   2 | 0,  2,  1,  3,  2,  2,  1,  4,  3,  2,  1,  3,  2,  2,  1,  5,  4,
   3 | 0,  3,  1,  4,  2,  3,  1,  5,  3,  3,  1,  4,  2,  3,  1,  6,  4,
   4 | 0,  4,  1,  5,  2,  4,  1,  6,  3,  4,  1,  5,  2,  4,  1,  7,  4,
   5 | 0,  5,  1,  6,  2,  5,  1,  7,  3,  5,  1,  6,  2,  5,  1,  8,  4,
   6 | 0,  6,  1,  7,  2,  6,  1,  8,  3,  6,  1,  7,  2,  6,  1,  9,  4,
   7 | 0,  7,  1,  8,  2,  7,  1,  9,  3,  7,  1,  8,  2,  7,  1, 10,  4,
   8 | 0,  8,  1,  9,  2,  8,  1, 10,  3,  8,  1,  9,  2,  8,  1, 11,  4,
   9 | 0,  9,  1, 10,  2,  9,  1, 11,  3,  9,  1, 10,  2,  9,  1, 12,  4,
  10 | 0, 10,  1, 11,  2, 10,  1, 12,  3, 10,  1, 11,  2, 10,  1, 13,  4,
  11 | 0, 11,  1, 12,  2, 11,  1, 13,  3, 11,  1, 12,  2, 11,  1, 14,  4,
  12 | 0, 12,  1, 13,  2, 12,  1, 14,  3, 12,  1, 13,  2, 12,  1, 15,  4,
  13 | 0, 13,  1, 14,  2, 13,  1, 15,  3, 13,  1, 14,  2, 13,  1, 16,  4,
  14 | 0, 14,  1, 15,  2, 14,  1, 16,  3, 14,  1, 15,  2, 14,  1, 17,  4,
  15 | 0, 15,  1, 16,  2, 15,  1, 17,  3, 15,  1, 16,  2, 15,  1, 18,  4,
  16 | 0, 16,  1, 17,  2, 16,  1, 18,  3, 16,  1, 17,  2, 16,  1, 19,  4,
  17 | 0, 17,  1, 18,  2, 17,  1, 19,  3, 17,  1, 18,  2, 17,  1, 20,  4,
etc.
A000225(4)^4 = ((2^4)-1)^4 = 50625 and A007088(50625) = "1100010111000001", where the rightmost run of 0-bits has length 5, therefore A(4,4) = 5.
A000225(3)^5 = ((2^3)-1)^5 = 16807 and A007088(16807) = "100000110100111", where the rightmost run of 0-bits has length 2, therefore A(3,5) = 2.
A000225(5)^3 = ((2^5)-1)^3 = 29791 and A007088(29791) = "111010001011111", where the rightmost run of 0-bits is a singleton, therefore A(5,3) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..22155
Index entries for sequences related to binary expansion of n

Cf. A000225, A001511, A007088, A007814, A285097.

A285097[n_]:=If[DigitCount[n,2,1]<2,0,IntegerExponent[BitAnd[n-1,n],2]-IntegerExponent[n,2]];A366370[n_,k_]:=A285097[1+(2^n-1)^k];
Table[A366370[k,n-k+1],{n,20},{k,n}] (* Paolo Xausa, Dec 02 2023 *)

up_to = 105;
A285097(n) = if(!n || !bitand(n,n-1), 0, valuation((n>>valuation(n,2))-1, 2));
A366370sq(n,k) = A285097(1+(((2^n)-1)^k));
\\ Or more directly as:
A366370sq(n,k) = if(1==n||1==k, 0, if(!(k%2), n, 1)+valuation(k>>1,2));
A366370list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A366370sq(col,(a-(col-1))))); (v); };
v366370 = A366370list(up_to);
A366370(n) = v366370[n];

A(n,k) = A285097(1+(A000225(n)^k)).

For all n >= 2, k >= 2, A(n,2k) = n+A007814(k), A(n,2k+1) = 1+A007814(k).

cp's OEIS Frontend

A285099 a(n) is the zero-based index of the second least significant 1-bit in the base-2 representation of n, or 0 if there are fewer than two 1-bits in n.

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