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%I #5 May 15 2024 16:29:02
%S 6,7,10,11,13,14,18,19,22,23,24,25,26,27,28,30,31,33,34,35,37,38,39,
%T 40,41,44,49,50,52,56,57,58,62,69,70,72,74,75,76,77,82,83,85,86,87,88,
%U 90,92,96,98,100,102,103,104,106,107,108,109,112,117,120,123
%N Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is even.
%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 odd version is A372586.
%F Numbers k such that A029931(k) + A056239(k) is even.
%e The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
%e {2,3} 6 (2,1)
%e {1,2,3} 7 (4)
%e {2,4} 10 (3,1)
%e {1,2,4} 11 (5)
%e {1,3,4} 13 (6)
%e {2,3,4} 14 (4,1)
%e {2,5} 18 (2,2,1)
%e {1,2,5} 19 (8)
%e {2,3,5} 22 (5,1)
%e {1,2,3,5} 23 (9)
%e {4,5} 24 (2,1,1,1)
%e {1,4,5} 25 (3,3)
%e {2,4,5} 26 (6,1)
%e {1,2,4,5} 27 (2,2,2)
%e {3,4,5} 28 (4,1,1)
%e {2,3,4,5} 30 (3,2,1)
%e {1,2,3,4,5} 31 (11)
%e {1,6} 33 (5,2)
%e {2,6} 34 (7,1)
%e {1,2,6} 35 (4,3)
%e {1,3,6} 37 (12)
%e {2,3,6} 38 (8,1)
%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],EvenQ[Total[bix[#]]+Total[prix[#]]]&]
%Y Positions of even terms in A372428, zeros A372427.
%Y For minimum (A372437) we have A372440, complement A372439.
%Y For length (A372441, zeros A071814) we have A372591, complement A372590.
%Y For maximum (A372442, zeros A372436) we have A372589, complement A372588.
%Y The complement is A372586.
%Y For just binary indices:
%Y - length: A001969, complement A000069
%Y - sum: A158704, complement A158705
%Y - minimum: A036554, complement A003159
%Y - maximum: A053754, complement A053738
%Y For just prime indices:
%Y - length: A026424 A028260 (count A027187), complement (count A027193)
%Y - sum: A300061 (count A058696), complement A300063 (count A058695)
%Y - minimum: A340933 (count A026805), complement A340932 (count A026804)
%Y - maximum: A244990 (count A027187), complement A244991 (count A027193)
%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, A066207, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.
%K nonn
%O 1,1
%A _Gus Wiseman_, May 14 2024
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