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A237287 Numbers that are not practical: positive integers n such that there exists at least one number k <= sigma(n) that is not a sum of distinct divisors of n.

%I A237287 #24 Dec 15 2024 06:46:57
%S A237287 3,5,7,9,10,11,13,14,15,17,19,21,22,23,25,26,27,29,31,33,34,35,37,38,
%T A237287 39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,62,63,65,67,68,69,
%U A237287 70,71,73,74,75,76,77,79,81,82,83,85,86,87,89,91,92,93,94,95,97,98,99,101,102,103,105
%N A237287 Numbers that are not practical: positive integers n such that there exists at least one number k <= sigma(n) that is not a sum of distinct divisors of n.
%C A237287 Complement of A005153 (practical numbers).
%C A237287 Numbers n such that A030057(n) < n.
%C A237287 First differs from A237046 at a(48).
%C A237287 First differs from A238524 at a(55). - _Omar E. Pol_, Mar 09 2014
%C A237287 First differs from A378471 at a(72). - _Hartmut F. W. Hoft_, Nov 27 2024
%H A237287 Jaroslav Krizek, <a href="/A237287/b237287.txt">Table of n, a(n) for n = 1..5000</a>
%e A237287 5 is in the sequence because there are 3 numbers <= sigma(5) = 6 that are not a sum of any subset of distinct divisors of 5: 2, 3 and 4.
%o A237287 (Python)
%o A237287 from itertools import count, islice
%o A237287 from sympy import factorint
%o A237287 def A237287_gen(startvalue=1): # generator of terms
%o A237287     for m in count(max(startvalue,1)):
%o A237287         if m > 1:
%o A237287             l = (~m & m-1).bit_length()
%o A237287             if l>0:
%o A237287                 P = (1<<l+1)-1
%o A237287                 for p, e in factorint(m>>l).items():
%o A237287                     if p > 1+P:
%o A237287                         yield m
%o A237287                         break
%o A237287                     P *= (p**(e+1)-1)//(p-1)
%o A237287             else:
%o A237287                 yield m
%o A237287 A237387_list = list(islice(A237287_gen(),30)) # _Chai Wah Wu_, Jul 05 2023
%Y A237287 Cf. A000203, A005153, A237046, A237289, A237290, A378471.
%K A237287 nonn
%O A237287 1,1
%A A237287 _Jaroslav Krizek_, Mar 02 2014
%E A237287 More terms added by _Hartmut F. W. Hoft_, Nov 27 2024, in order to show the difference from A378471.