cp's OEIS Frontend

A372590 Numbers whose binary weight (A000120) plus bigomega (A001222) is odd.

%I A372590 #7 May 15 2024 16:28:50
%S A372590 1,3,4,5,12,14,16,17,18,20,21,22,23,25,26,27,29,30,35,38,43,45,48,49,
%T A372590 53,55,56,62,63,64,66,68,69,71,72,74,75,78,80,81,82,83,84,87,88,89,91,
%U A372590 92,93,94,99,100,101,102,104,105,108,113,114,115,116,118,120
%N A372590 Numbers whose binary weight (A000120) plus bigomega (A001222) is odd.
%C A372590 The even version is A372591.
%e A372590 The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
%e A372590         {1}   1  ()
%e A372590       {1,2}   3  (2)
%e A372590         {3}   4  (1,1)
%e A372590       {1,3}   5  (3)
%e A372590       {3,4}  12  (2,1,1)
%e A372590     {2,3,4}  14  (4,1)
%e A372590         {5}  16  (1,1,1,1)
%e A372590       {1,5}  17  (7)
%e A372590       {2,5}  18  (2,2,1)
%e A372590       {3,5}  20  (3,1,1)
%e A372590     {1,3,5}  21  (4,2)
%e A372590     {2,3,5}  22  (5,1)
%e A372590   {1,2,3,5}  23  (9)
%e A372590     {1,4,5}  25  (3,3)
%e A372590     {2,4,5}  26  (6,1)
%e A372590   {1,2,4,5}  27  (2,2,2)
%e A372590   {1,3,4,5}  29  (10)
%e A372590   {2,3,4,5}  30  (3,2,1)
%e A372590     {1,2,6}  35  (4,3)
%e A372590     {2,3,6}  38  (8,1)
%e A372590   {1,2,4,6}  43  (14)
%e A372590   {1,3,4,6}  45  (3,2,2)
%t A372590 Select[Range[100],OddQ[DigitCount[#,2,1]+PrimeOmega[#]]&]
%Y A372590 For sum (A372428, zeros A372427) we have A372586, complement A372587.
%Y A372590 For minimum (A372437) we have A372439, complement A372440.
%Y A372590 Positions of odd terms in A372441, zeros A071814.
%Y A372590 For maximum (A372442, zeros A372436) we have A372588, complement A372589.
%Y A372590 The complement is A372591.
%Y A372590 For just binary indices:
%Y A372590 - length: A000069, complement A001969
%Y A372590 - sum: A158705, complement A158704
%Y A372590 - minimum: A003159, complement A036554
%Y A372590 - maximum: A053738, complement A053754
%Y A372590 For just prime indices:
%Y A372590 - length: A026424 (count A027193), complement A028260 (count A027187)
%Y A372590 - sum: A300063 (count A058695), complement A300061 (count A058696)
%Y A372590 - minimum: A340932 (count A026804), complement A340933 (count A026805)
%Y A372590 - maximum: A244991 (count A027193), complement A244990 (count A027187)
%Y A372590 A005408 lists odd numbers.
%Y A372590 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372590 A029837 gives greatest binary index, least A001511.
%Y A372590 A031368 lists odd-indexed primes, even A031215.
%Y A372590 A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A372590 A070939 gives length of binary expansion.
%Y A372590 A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
%Y A372590 Cf. A000720, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.
%K A372590 nonn
%O A372590 1,2
%A A372590 _Gus Wiseman_, May 14 2024