A394987 a(n) is the least product x*y of two integers 2^(n-1) <= x <= y < 2^n maximizing the Hamming weight of this product.

1, 6, 30, 126, 510, 2046, 7935, 32767, 196607, 524286, 2031615, 8323071, 50331647, 134217723, 535822335, 2143289343, 8573157375, 66571993087, 103079215103, 549755813887, 2164663517183, 8796093022207, 35184372088831, 140737488355327, 558551906910207, 2250700302057471

Offset

1,2

Links

Bert Dobbelaere, Table of n, a(n) for n = 1..100

Formula

a(n) = A394989(n)*A394990(n).

Example

   n  A394988(n)=A000120(a(n))         |
          x=A394989(n)                 |   x=A395003(n)
                 y=A394990(n)          |          y=A395004(n)
                              x*y=a(n) |                 x*y=A394991(n)
                                       |
   1  1       1       1              1 |       1       1              1
   2  2       2       3              6 |       3       3              9
   3  4       5       6             30 |       5       6             30
   4  6       9      14            126 |       9      14            126
   5  8      17      30            510 |      17      30            510
   6 10      33      62           2046 |      33      62           2046
   7 12      69     115           7935 |     105     117          12285
   8 15     151     217          32767 |     151     217          32767
   9 17     421     467         196607 |     421     467         196607
  10 18     513    1022         524286 |     933     979         913407
  11 20    1341    1515        2031615 |    1705    1845        3145725
  12 22    2391    3481        8323071 |    3767    3897       14679999
  13 25    6563    7669       50331647 |    7535    7793       58720255
  14 26   10209   13147      134217723 |   15727   15985      251396095
  15 28   23061   23235      535822335 |   31497   31943     1006108671
  16 30   42069   50947     2143289343 |   64265   64711     4158652415
  17 32   66225  129455     8573157375 |  128751  129265    16642998015
  18 35  257503  258529    66571993087 |  257503  258529    66571993087
  19 36  296323  347861   103079215103 |  519647  520673   270566162431
  20 39  647089  849583   549755813887 |  647089  849583   549755813887

Prog

(Python)
from sympy.utilities.iterables import multiset_permutations
from sympy import divisors
def A394987(n):
    a = 1<<n-1
    b = a<<1
    k = (n<<1)-1
    for l in range(k, 0, -1):
        for s in multiset_permutations('0'*(k+1-l)+'1'*l):
            m = int(''.join(s), 2)
            for d in divisors(m):
                if d**2>m:
                    break
                if a<=d<b and a*d<=m<b*d:
                    return m # Chai Wah Wu, Apr 11 2026

Crossrefs

Cf. A000120, A209862, A230097, A357304, A394988, A394989, A394990, A394991, A395003, A395004.

Sequence in context: A366058 A356835 A344344 * A002446 A002934 A174319

Adjacent sequences: A394984 A394985 A394986 * A394988 A394989 A394990

Keyword

nonn

Author

Hugo Pfoertner, Apr 09 2026

Status

approved