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A295827 a(n) = least odd k > 1 such that n and n*k have the same Hamming weight, or -1 if no such k exists. 1
-1, -1, 3, -1, 13, 3, 3, -1, 57, 13, 35, 3, 21, 3, 3, -1, 241, 57, 7, 13, 13, 35, 39, 3, 169, 21, 5, 3, 21, 3, 3, -1, 993, 241, 11, 57, 7, 7, 5, 13, 3197, 13, 9, 35, 3, 39, 13, 3, 21, 169, 3, 21, 39, 5, 47, 3, 27, 21, 5, 3, 13, 3, 3, -1, 4033, 993, 491, 241 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The Hamming weight of a number n is given by A000120(n).

Apparently, a(n) = -1 iff n = 2^k for some k >= 0.

Apparently, a(2^n + 1) = A020515(n) for any n > 1.

a(2^n - 1) = 3 for any n > 1.

a(n) = 3 iff n = A077459(k) for some k > 1.

This sequence has similarities with A292849: here we want A000120(n*a(n)) = A000120(n), there we want A000120(n*a(n)) = A000120(a(n)).

For any n > 0, if a(n) > 0 then A292849(a(n)) <= n.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..8192

Rémy Sigrist, Logarithmic scatterplot of the sequence for n=1..2^17 and a(n) < 10^18

FORMULA

a(2*n) = a(n) for any n > 0.

EXAMPLE

The first terms, alongside the binary representations of n and of n*a(n), are:

  n     a(n)     bin(n)         bin(n*a(n))

  --    ----     ------         -----------

   1      -1          1                  -1

   2      -1         10                 -10

   3       3         11                1001

   4      -1        100                -100

   5      13        101             1000001

   6       3        110               10010

   7       3        111               10101

   8      -1       1000               -1000

   9      57       1001          1000000001

  10      13       1010            10000010

  11      35       1011           110000001

  12       3       1100              100100

  13      21       1101           100010001

  14       3       1110              101010

  15       3       1111              101101

  16      -1      10000              -10000

  17     241      10001       1000000000001

  18      57      10010         10000000010

  19       7      10011            10000101

  20      13      10100           100000100

MAPLE

f:= proc(n) local k, w;

  if n = 2^padic:-ordp(n, 2) then return -1 fi;

  w:= convert(convert(n, base, 2), `+`);

  for k from 3 by 2 do

    if convert(convert(n*k, base, 2), `+`)=w then return k fi

  od

end proc:

map(f, [$1..100]); # Robert Israel, Nov 28 2017

MATHEMATICA

Table[SelectFirst[Range[3, 10^4 + 1, 2], SameQ @@ Map[DigitCount[#, 2, 1] &, {n, n #}] &] /. m_ /; MissingQ@ m -> -1, {n, 68}] (* Michael De Vlieger, Nov 28 2017 *)

PROG

(PARI) A057168(n)=n+bitxor(n, n+n=bitand(n, -n))\n\4+n \\ after M. F. Hasler at A057168

a(n) = n\=2^valuation(n, 2); if (n==1, -1, my(w=(n-1)/2); while(1, w=A057168(w); if((2*w+1)%n==0, return((2*w+1)/n))))

CROSSREFS

Cf. A000120, A020515, A057168, A077459, A292849.

Sequence in context: A248843 A170910 A134768 * A277197 A297898 A322384

Adjacent sequences:  A295824 A295825 A295826 * A295828 A295829 A295830

KEYWORD

sign,base

AUTHOR

Rémy Sigrist, Nov 28 2017

STATUS

approved

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Last modified November 19 08:44 EST 2019. Contains 329318 sequences. (Running on oeis4.)