A372587 - OEIS

%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