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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.
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%I #12 Mar 07 2023 19:04:19

%S 2,4,6,8,10,12,13,14,16,18,20,22,24,25,26,28,29,30,32,34,36,38,40,41,

%T 42,44,46,48,49,50,52,53,54,56,57,58,59,60,61,62,64,66,68,72,74,76,80,

%U 81,82,84,86,88,89,90,92,94,96,97,98,100,101,102,104,105,106

%N 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.

%C 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.

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

%e The initial terms, binary expansions, and positions of 1's are:

%e 2: 10 ~ {2}

%e 4: 100 ~ {3}

%e 6: 110 ~ {2,3}

%e 8: 1000 ~ {4}

%e 10: 1010 ~ {2,4}

%e 12: 1100 ~ {3,4}

%e 13: 1101 ~ {1,3,4}

%e 14: 1110 ~ {2,3,4}

%e 16: 10000 ~ {5}

%e 18: 10010 ~ {2,5}

%e 20: 10100 ~ {3,5}

%e 22: 10110 ~ {2,3,5}

%e 24: 11000 ~ {4,5}

%e 25: 11001 ~ {1,4,5}

%e 26: 11010 ~ {2,4,5}

%e 28: 11100 ~ {3,4,5}

%e 29: 11101 ~ {1,3,4,5}

%e 30: 11110 ~ {2,3,4,5}

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

%o (Python 3.10+)

%o from itertools import count, islice

%o def A359496_gen(startvalue=0): # generator of terms >= startvalue

%o 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)))

%o A359496_list = list(islice(A359496_gen(),20)) # _Chai Wah Wu_, Jan 19 2023

%Y The opposite version is A359401.

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

%Y A030190 gives binary expansion, reverse A030308.

%Y A070939 counts binary digits.

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

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

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

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

%K nonn,base

%O 1,1

%A _Gus Wiseman_, Jan 18 2023