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A386294 Nonsquarefree weak numbers k such that A053669(k) < A006530(k).

%I A386294 #6 Jul 21 2025 13:14:46
%S A386294 20,28,40,44,45,50,52,56,63,68,75,76,80,84,88,92,98,99,104,112,116,
%T A386294 117,124,126,132,135,136,140,147,148,152,153,156,160,164,168,171,172,
%U A386294 175,176,184,188,189,198,204,207,208,212,220,224,228,232,234,236,242,244
%N A386294 Nonsquarefree weak numbers k such that A053669(k) < A006530(k).
%H A386294 Michael De Vlieger, <a href="/A386294/b386294.txt">Table of n, a(n) for n = 1..10000</a>
%F A386294 Intersection of A332785 and A080259 = A332785 \ A055932 = A126706 \ A286708 \ A380543.
%e A386294 Table of n, a(n) and prime decomposition for n = 1..12:
%e A386294  n  a(n)
%e A386294 ------------------
%e A386294  1   20 = 2^2 * 5
%e A386294  2   28 = 2^2 * 7
%e A386294  3   40 = 2^3 * 5
%e A386294  4   44 = 2^2 * 11
%e A386294  5   45 = 3^2 * 5
%e A386294  6   50 = 2 * 5^2
%e A386294  7   52 = 2^2 * 13
%e A386294  8   56 = 2^3 * 7
%e A386294  9   63 = 3^2 * 7
%e A386294 10   68 = 2^2 * 17
%e A386294 11   75 = 3 * 5^2
%e A386294 12   76 = 2^2 * 19
%e A386294 Let q = A053669 and let gpf = A006530.
%e A386294 The number 12 = 2^2*3 is not in the sequence since q(12) > gpf(12), i.e., 5 > 3.
%e A386294 The number 18 = 2*3^2 is not in the sequence since q(18) > gpf(18), i.e., 5 > 3.
%e A386294 a(1) = 20 = 2^2*5 since q(20) < gpf(20), i.e., 3 < 5.
%e A386294 The number 60 = 2^2*3*5 is not a term since q(60) > gpf(60), i.e., 7 > 5, etc.
%t A386294 f[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q]]; q]; Select[Range[256], Nor[Length[#2] == 1, Max[#2[[All, -1]]] == 1, Divisible[#1, Apply[Times, #2[[All, 1]]]^2], f[#1] > #2[[-1, 1]]] & @@ {#, FactorInteger[#]} &]
%Y A386294 Cf. A001694, A002110, A006939, A007947, A013929, A052485, A053669, A055932, A080259, A126706, A286708, A332785, A380543.
%K A386294 nonn
%O A386294 1,1
%A A386294 _Michael De Vlieger_, Jul 19 2025