A359496 Nonnegative integers whose sum of positions of 1's in their binary expansion is less than the sum of positions of 1's in their reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

2, 4, 6, 8, 10, 12, 13, 14, 16, 18, 20, 22, 24, 25, 26, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 44, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 64, 66, 68, 72, 74, 76, 80, 81, 82, 84, 86, 88, 89, 90, 92, 94, 96, 97, 98, 100, 101, 102, 104, 105, 106

Offset

1,1

Comments

First differs from A161602 in lacking 70, with binary expansion (1,0,0,0,1,1,0), positions of 1's 1 + 5 + 6 = 12, reversed 2 + 3 + 7 = 12.

Links

Table of n, a(n) for n=1..65.

Formula

A230877(a(n)) < A029931(a(n)).

Example

The initial terms, binary expansions, and positions of 1's are:
    2:      10 ~ {2}
    4:     100 ~ {3}
    6:     110 ~ {2,3}
    8:    1000 ~ {4}
   10:    1010 ~ {2,4}
   12:    1100 ~ {3,4}
   13:    1101 ~ {1,3,4}
   14:    1110 ~ {2,3,4}
   16:   10000 ~ {5}
   18:   10010 ~ {2,5}
   20:   10100 ~ {3,5}
   22:   10110 ~ {2,3,5}
   24:   11000 ~ {4,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}
   28:   11100 ~ {3,4,5}
   29:   11101 ~ {1,3,4,5}
   30:   11110 ~ {2,3,4,5}

Mathematica

Select[Range[100], Total[Accumulate[IntegerDigits[#, 2]]]>Total[Accumulate[Reverse[IntegerDigits[#, 2]]]]&]

Prog

(Python 3.10+)
from itertools import count, islice
def A359496_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:sum(i for i, j in enumerate(bin(n)[2:]) if j=='1')<<1 < n.bit_count()*(n.bit_length()-1), count(max(startvalue, 0)))
A359496_list = list(islice(A359496_gen(), 20)) # Chai Wah Wu, Jan 19 2023

Crossrefs

The opposite version is A359401.

Indices of negative terms in A359495; indices of 0's are A359402.

A030190 gives binary expansion, reverse A030308.

A070939 counts binary digits.

A230877 adds up positions of 1's in binary expansion, reverse A029931.

A326669 lists numbers with integer mean position of a 1 in binary expansion.

A358194 counts partitions by sum of partial sums, compositions A053632.

Cf. A000120, A048793, A051293, A222955, A231204, A291166, A304818, A326672, A326673, A359043.

Sequence in context: A365984 A301454 A161602 * A356066 A285591 A327210

Adjacent sequences: A359493 A359494 A359495 * A359497 A359498 A359499

Keyword

nonn,base

Author

Gus Wiseman, Jan 18 2023

Status

approved