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.

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

1,5

Links

Antti Karttunen, Table of n, a(n) for n = 1..22155

Index entries for sequences related to binary expansion of n

Formula

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).

Example

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.

Mathematica

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 *)

Prog

(PARI)
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];

Crossrefs

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

Sequence in context: A056885 A029373 A357645 * A297617 A351982 A029362

Adjacent sequences: A366367 A366368 A366369 * A366371 A366372 A366373

Keyword

nonn,tabl

Author

Antti Karttunen, Oct 14 2023

Status

approved