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A364061 Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.

%I A364061 #32 Mar 27 2025 23:27:51
%S A364061 2,4,8,16,18,32,50,54,64,98,108,128,162,242,250,256,324,338,450,486,
%T A364061 500,512,578,648,686,722,882,972,1024,1058,1250,1350,1372,1458,1682,
%U A364061 1922,1944,2048,2178,2250,2450,2500,2646,2662,2738,2916,3042,3362,3698,3888
%N A364061 Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.
%C A364061 Also numbers whose multiset of prime factors has unique co-mode 2. Here, a co-mode in a multiset is an element that appears at most as many times as each of the other elements. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.
%H A364061 Robert Israel, <a href="/A364061/b364061.txt">Table of n, a(n) for n = 1..10000</a>
%F A364061 Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=2} (1-1/2^(k-1))*(s(k)-s(k+1)) = 1.16896822653093929144..., where s(k) = Product_{primes p >= 3} (1 + 1/(p^(k-1)*(p-1))) is the sum of reciprocals of the odd k-full numbers (numbers whose prime factorization has no exponent that is smaller than k). - _Amiram Eldar_, Aug 30 2024
%e A364061 The terms together with their prime factors begin:
%e A364061     2 = 2
%e A364061     4 = 2*2
%e A364061     8 = 2*2*2
%e A364061    16 = 2*2*2*2
%e A364061    18 = 2*3*3
%e A364061    32 = 2*2*2*2*2
%e A364061    50 = 2*5*5
%e A364061    54 = 2*3*3*3
%e A364061    64 = 2*2*2*2*2*2
%e A364061    98 = 2*7*7
%e A364061   108 = 2*2*3*3*3
%e A364061   128 = 2*2*2*2*2*2*2
%p A364061 filter:= proc(n) local F,F2,Fo;
%p A364061   F:= ifactors(n)[2];
%p A364061   F2,Fo:= selectremove(t -> t[1]=2, F);
%p A364061   Fo = [] or F2[1,2] < min(Fo[..,2])
%p A364061 end proc:
%p A364061 select(filter, 2*[$1..5000]); # _Robert Israel_, Apr 22 2024
%t A364061 prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A364061 comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
%t A364061 Select[Range[100],comodes[prifacs[#]]=={2}&]
%o A364061 (Python)
%o A364061 from sympy import factorint
%o A364061 from itertools import count, islice
%o A364061 def A364061_gen(startvalue=2): # generator of terms >= startvalue
%o A364061     return filter(lambda n:(l:=(~n&n-1).bit_length()) < min(factorint(m:=n>>l).values(),default=0) or m==1, count(max(startvalue+startvalue&1,2),2))
%o A364061 A364061_list = list(islice(A364061_gen(),30)) # _Chai Wah Wu_, Jul 14 2023
%Y A364061 For any unique co-mode: A359178, counted by A362610, complement A362606.
%Y A364061 For high mode: A360013, positions of 1's in A363487, counted by A241131.
%Y A364061 For low mode: A360015, positions of 1's in A363486, counted by A241131.
%Y A364061 Partitions of this type are counted by A364062.
%Y A364061 For low co-mode: A364158, positions of 1's in A364192, counted by A364159.
%Y A364061 Positions of 1's in A364191, high A364192.
%Y A364061 A112798 lists prime indices, length A001222, sum A056239.
%Y A364061 A356862 ranks partitions w/ unique mode, count A362608, complement A362605.
%Y A364061 A362611 counts modes in prime indices, triangle A362614.
%Y A364061 A362613 counts co-modes in prime indices, triangle A362615.
%Y A364061 Cf. A000265, A007814, A327473, A327476, A362616, A360014, A363722, A363723, A363725, A363727, A363730.
%K A364061 nonn
%O A364061 1,1
%A A364061 _Gus Wiseman_, Jul 12 2023