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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.
0
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.
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.
Sequence in context: A365984 A301454 A161602 * A356066 A285591 A327210
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Jan 18 2023
STATUS
approved