A372586 - OEIS

%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