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A379097 Numbers that are not waterproof.

%I A379097 #13 Dec 20 2024 12:44:34
%S A379097 60,84,120,132,156,168,204,228,240,264,276,280,300,312,315,336,348,
%T A379097 372,408,420,440,444,456,480,492,495,516,520,528,552,560,564,585,588,
%U A379097 600,616,624,630,636,660,672,680,693,696,708,728,732,744,760,765,780,804,816
%N A379097 Numbers that are not waterproof.
%C A379097 Zero and one are waterproof numbers by convention. Numbers that admit a prime factorization are not waterproof if their water capacity is > 0. (The water capacity of a number is defined in A275339.)
%C A379097 Proper subset of A375055, in turn a proper subset of A126706, since A001221(a(n)) >= 3 and a maximum multiplicity is required for at least one prime power factor, so as to have positive water capacity. - _Michael De Vlieger_, Dec 18 2024
%H A379097 Michael De Vlieger, <a href="/A379097/b379097.txt">Table of n, a(n) for n = 1..10000</a>
%p A379097 # The function 'water_capacity' is defined in A275339.
%p A379097 is_not_waterproof := n -> ifelse(n < 2, false, is(water_capacity(n) <> 0)):
%p A379097 select(is_not_waterproof, [seq(0..820)]);
%t A379097 nn = 816;
%t A379097 s = Select[Range[nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
%t A379097 Select[s, Function[f, And[NoneTrue[{Sort[f], ReverseSort[f]}, # == f &],
%t A379097   Total[(f //. {a___, b_, c__, d_, e___} /;
%t A379097     AllTrue[{c}, And[# < b, # < d] &] :>
%t A379097     {a, b, Sequence @@ Table[Min[b, d], {Length[{c}]}], d, e}) - f] > 0] ]
%t A379097 [Power @@@ FactorInteger[#]] &] (* _Michael De Vlieger_, Dec 18 2024, after _Jean-François Alcover_ at A275339 *)
%o A379097 (Python)
%o A379097 # The function 'WaterCapacity' is defined in A275339.
%o A379097 print([n for n in range(818) if WaterCapacity(n) > 0])
%Y A379097 Cf. A275339, A379094, A379096, A379098.
%K A379097 nonn
%O A379097 1,1
%A A379097 _Peter Luschny_, Dec 16 2024