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%I #12 Feb 12 2023 03:14:02
%S 0,1,1,1,1,2,2,1,2,3,4,2,1,4,4,2,2,4,4,2,3,4,5,2,2,4,5,3,3,7,5,4,4,5,
%T 6,2,6,9,8,4,4,9,6,4,4,8,8,4,5,7,8,3,4,6,8,5,4,6,7,5,4,9,7,4,6,10,8,5,
%U 7,9,9,3,7,10,10,2,6,12,8,5,4,10,11,5,6,11,12,5,6,8,9,6,8,11,11,6
%N Number of ways to write n as sum of a powerful and a squarefree number.
%H T. D. Noe, <a href="/A085257/b085257.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerfulNumber.html">Powerful Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>.
%F a(n) > 0 for n > 1 (proof?).
%F a(A085258(n)) = n and a(i) <> n for i < A085258(n).
%e a(10) = 3: 10 = 1 + 9 = A005117(1) + A001694(4) = 2 + 8 = A005117(2) + A001694(3) = 6 + 4 = A005117(5) + A001694(2).
%t With[{m = 100}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; sqf = Select[Range[m], SquareFreeQ]; BinCounts[Select[Plus @@@ Tuples[{pow, sqf}], # <= m &], {1, m, 1}]] (* _Amiram Eldar_, Feb 12 2023 *)
%Y Cf. A001694, A005117, A085258.
%K nonn
%O 1,6
%A _Reinhard Zumkeller_, Jun 23 2003
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