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
Robert Israel, Table of n, a(n) for n = 1..10000
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))
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 12 2023
STATUS
approved