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A364061
Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.
9
2, 4, 8, 16, 18, 32, 50, 54, 64, 98, 108, 128, 162, 242, 250, 256, 324, 338, 450, 486, 500, 512, 578, 648, 686, 722, 882, 972, 1024, 1058, 1250, 1350, 1372, 1458, 1682, 1922, 1944, 2048, 2178, 2250, 2450, 2500, 2646, 2662, 2738, 2916, 3042, 3362, 3698, 3888
OFFSET
1,1
COMMENTS
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}.
LINKS
FORMULA
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
EXAMPLE
The terms together with their prime factors begin:
2 = 2
4 = 2*2
8 = 2*2*2
16 = 2*2*2*2
18 = 2*3*3
32 = 2*2*2*2*2
50 = 2*5*5
54 = 2*3*3*3
64 = 2*2*2*2*2*2
98 = 2*7*7
108 = 2*2*3*3*3
128 = 2*2*2*2*2*2*2
MAPLE
filter:= proc(n) local F, F2, Fo;
F:= ifactors(n)[2];
F2, Fo:= selectremove(t -> t[1]=2, F);
Fo = [] or F2[1, 2] < min(Fo[.., 2])
end proc:
select(filter, 2*[$1..5000]); # Robert Israel, Apr 22 2024
MATHEMATICA
prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
comodes[ms_]:=Select[Union[ms], Count[ms, #]<=Min@@Length/@Split[ms]&];
Select[Range[100], comodes[prifacs[#]]=={2}&]
PROG
(Python)
from sympy import factorint
from itertools import count, islice
def A364061_gen(startvalue=2): # generator of terms >= startvalue
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))
A364061_list = list(islice(A364061_gen(), 30)) # Chai Wah Wu, Jul 14 2023
CROSSREFS
For any unique co-mode: A359178, counted by A362610, complement A362606.
For high mode: A360013, positions of 1's in A363487, counted by A241131.
For low mode: A360015, positions of 1's in A363486, counted by A241131.
Partitions of this type are counted by A364062.
For low co-mode: A364158, positions of 1's in A364192, counted by A364159.
Positions of 1's in A364191, high A364192.
A112798 lists prime indices, length A001222, sum A056239.
A356862 ranks partitions w/ unique mode, count A362608, complement A362605.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
Sequence in context: A088827 A316900 A076057 * A133809 A128700 A331579
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 12 2023
STATUS
approved