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%I #8 May 15 2024 16:29:08
%S 1,2,3,4,5,8,9,12,15,16,17,20,21,29,32,36,42,43,45,46,47,48,51,53,54,
%T 55,59,60,61,63,64,65,66,67,68,71,73,78,79,80,81,84,89,91,93,94,95,97,
%U 99,101,105,110,111,113,114,115,116,118,119,121,122,125,127
%N Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is odd.
%C A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C The even version is A372587.
%F Numbers k such that A029931(k) + A056239(k) is odd.
%e The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
%e {1} 1 ()
%e {2} 2 (1)
%e {1,2} 3 (2)
%e {3} 4 (1,1)
%e {1,3} 5 (3)
%e {4} 8 (1,1,1)
%e {1,4} 9 (2,2)
%e {3,4} 12 (2,1,1)
%e {1,2,3,4} 15 (3,2)
%e {5} 16 (1,1,1,1)
%e {1,5} 17 (7)
%e {3,5} 20 (3,1,1)
%e {1,3,5} 21 (4,2)
%e {1,3,4,5} 29 (10)
%e {6} 32 (1,1,1,1,1)
%e {3,6} 36 (2,2,1,1)
%e {2,4,6} 42 (4,2,1)
%e {1,2,4,6} 43 (14)
%e {1,3,4,6} 45 (3,2,2)
%e {2,3,4,6} 46 (9,1)
%e {1,2,3,4,6} 47 (15)
%e {5,6} 48 (2,1,1,1,1)
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Select[Range[100],OddQ[Total[bix[#]]+Total[prix[#]]]&]
%Y Positions of odd terms in A372428, zeros A372427.
%Y For minimum (A372437) we have A372439, complement A372440.
%Y For length (A372441, zeros A071814) we have A372590, complement A372591.
%Y For maximum (A372442, zeros A372436) we have A372588, complement A372589.
%Y The complement is A372587.
%Y For just binary indices:
%Y - length: A000069, complement A001969
%Y - sum: A158705, complement A158704
%Y - minimum: A003159, complement A036554
%Y - maximum: A053738, complement A053754
%Y For just prime indices:
%Y - length: A026424 (count A027193), complement A028260 (count A027187)
%Y - sum: A300063 (count A058695), complement A300061 (count A058696)
%Y - minimum: A340932 (count A026804), complement A340933 (count A026805)
%Y - maximum: A244991 (count A027193), complement A244990 (count A027187)
%Y A005408 lists odd numbers.
%Y A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A029837 gives greatest binary index, least A001511.
%Y A031368 lists odd-indexed primes, even A031215.
%Y A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A061395 gives greatest prime index, least A055396.
%Y A070939 gives length of binary expansion.
%Y A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
%Y Cf. A000720, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 14 2024