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A385755 Numbers k with a unique combination of bigomega(k) and sopfr(k).

%I A385755 #31 Jul 13 2025 12:38:58
%S A385755 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,22,23,24,26,29,30,31,
%T A385755 32,34,35,36,37,38,41,43,46,47,48,53,58,59,61,62,64,67,70,71,72,73,74,
%U A385755 79,82,83,86,89,94,96,97,101,103,106,107,109,113,118,122,127,128,131
%N A385755 Numbers k with a unique combination of bigomega(k) and sopfr(k).
%H A385755 Michael De Vlieger, <a href="/A385755/b385755.txt">Table of n, a(n) for n = 1..10000</a>
%H A385755 Michael De Vlieger, <a href="/A385755/a385755.txt">Mathematica code</a>.
%H A385755 Michael De Vlieger, <a href="/A385755/a385755.png">Plot a(n) at (x,y) = (A001222(a(n)), A001414(a(n)))</a> for x <= 16 and y <= 33.
%e A385755 All primes p are in the sequence, because they are characterized by the pair [b,s] = [bigomega=1, sopfr=p], and no other numbers have this pair.
%e A385755 All even semiprimes 2*p are terms, because no other number can have [b,s]=[2,p+2]. p+2 is odd, and odd semiprimes p*q would have even s.
%e A385755 20 with [b,s]=[3,2+2+5] and 27 with [b,s]=[3,3+3+3] are not in the sequence, because both have [b,s]=[3,9].
%e A385755 21 and 25 are not in the sequence, because both have [b,s]=[2,10].
%e A385755 36 is in the sequence as it is the only number having [4, 10]. - _David A. Corneth_, Jul 11 2025
%e A385755 From _Michael De Vlieger_, Jul 13 2025: (Start)
%e A385755 Plot a(n) at (x,y) = (A001222(a(n)), A001414(a(n))):
%e A385755      0    1    2    3    4    5    6    7     8     9
%e A385755 -----------------------------------------------------
%e A385755  0:  1
%e A385755  1:
%e A385755  2:       2
%e A385755  3:       3
%e A385755  4:            4
%e A385755  5:       5    6
%e A385755  6:            9    8
%e A385755  7:       7   10   12
%e A385755  8:           15   18   16
%e A385755  9:           14        24
%e A385755 10:                30   36   32
%e A385755 11:      11                  48
%e A385755 12:           35             72   64
%e A385755 13:      13   22                  96
%e A385755 14:                70            144  128
%e A385755 15:           26                      192
%e A385755 16:                                   288   256
%e A385755 17:      17                                 384
%e A385755 18:                                         576   512
%e A385755 19:      19   34                                  768
%e A385755          ...  (End)
%Y A385755 Cf. A001222, A001414, A385756, A385811.
%Y A385755 Subsequences: A000040, A000079, A100484.
%K A385755 nonn
%O A385755 1,2
%A A385755 _Hugo Pfoertner_, Jul 09 2025