cp's OEIS Frontend

A372589 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is even.

%I A372589 #6 May 15 2024 16:28:55
%S A372589 3,4,5,9,12,13,14,16,17,20,22,23,25,30,31,35,36,37,38,39,42,43,48,49,
%T A372589 52,53,54,56,57,58,61,63,64,66,67,68,69,73,75,77,80,82,83,85,88,90,92,
%U A372589 93,94,97,99,100,102,103,109,110,115,118,119,120,121,123,124
%N A372589 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is even.
%C A372589 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 A372589 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 A372589 The odd version is A372588.
%F A372589 Numbers k such that A070939(k) + A061395(k) is even.
%e A372589 The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
%e A372589         {1,2}   3  (2)
%e A372589           {3}   4  (1,1)
%e A372589         {1,3}   5  (3)
%e A372589         {1,4}   9  (2,2)
%e A372589         {3,4}  12  (2,1,1)
%e A372589       {1,3,4}  13  (6)
%e A372589       {2,3,4}  14  (4,1)
%e A372589           {5}  16  (1,1,1,1)
%e A372589         {1,5}  17  (7)
%e A372589         {3,5}  20  (3,1,1)
%e A372589       {2,3,5}  22  (5,1)
%e A372589     {1,2,3,5}  23  (9)
%e A372589       {1,4,5}  25  (3,3)
%e A372589     {2,3,4,5}  30  (3,2,1)
%e A372589   {1,2,3,4,5}  31  (11)
%e A372589       {1,2,6}  35  (4,3)
%e A372589         {3,6}  36  (2,2,1,1)
%e A372589       {1,3,6}  37  (12)
%e A372589       {2,3,6}  38  (8,1)
%e A372589     {1,2,3,6}  39  (6,2)
%e A372589       {2,4,6}  42  (4,2,1)
%e A372589     {1,2,4,6}  43  (14)
%t A372589 Select[Range[2,100],EvenQ[IntegerLength[#,2]+PrimePi[FactorInteger[#][[-1,1]]]]&]
%Y A372589 For sum (A372428, zeros A372427) we have A372587, complement A372586.
%Y A372589 For minimum (A372437) we have A372440, complement A372439.
%Y A372589 For length (A372441, zeros A071814) we have A372591, complement A372590.
%Y A372589 Positions of even terms in A372442, zeros A372436.
%Y A372589 The complement is A372588.
%Y A372589 For just binary indices:
%Y A372589 - length: A001969, complement A000069
%Y A372589 - sum: A158704, complement A158705
%Y A372589 - minimum:  A036554, complement A003159
%Y A372589 - maximum: A053754, complement A053738
%Y A372589 For just prime indices:
%Y A372589 - length: A026424 A028260 (count A027187), complement (count A027193)
%Y A372589 - sum: A300061 (count A058696), complement A300063 (count A058695)
%Y A372589 - minimum: A340933 (count A026805), complement A340932 (count A026804)
%Y A372589 - maximum: A244990 (count A027187), complement A244991 (count A027193)
%Y A372589 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372589 A029837 gives greatest binary index, least A001511.
%Y A372589 A031215 lists even-indexed primes, odd A031368.
%Y A372589 A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A372589 A061395 gives greatest prime index, least A055396.
%Y A372589 A070939 gives length of binary expansion.
%Y A372589 A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
%Y A372589 Cf. A000720, A006141, A066207, A243055, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.
%K A372589 nonn
%O A372589 1,1
%A A372589 _Gus Wiseman_, May 14 2024