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 A216259 Numbers k that are not squarefree such that the difference between sigma(k) and usigma(k) is a square > 0. 1
 18, 28, 40, 54, 68, 84, 99, 120, 124, 184, 204, 208, 220, 284, 297, 315, 372, 388, 423, 424, 475, 508, 552, 616, 624, 660, 765, 796, 852, 900, 928, 940, 945, 963, 964, 1012, 1152, 1164, 1192, 1269, 1272, 1348, 1395, 1425, 1449, 1458, 1496, 1524, 1664, 1719, 1796, 1848, 1975 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS This sequence is a subsequence of A013929 (nonsquarefree numbers). If n were a squarefree number (A005117), then the difference would be 0. It appears that the terms are of the form q*p^q,  p prime. The corresponding squares are: 9, 16, 36, 36, 36, 64, 36, 144, 64, 144, 144, 196, 144,.... LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 MAPLE with(numtheory): for n from 1 to 2000 do :it:=1:s:=0:x:=divisors(n): n1:=nops(x): for k from 1 to n1 do:d:=x[k]:if gcd(d, n/d)=1 then s:=s+d:else fi:od: s1:=sigma(n): if sqrt(s1-s)=floor(sqrt(s1-s)) and s1>s then printf(`%d, `, n):else fi:od: MATHEMATICA lst={}; usigma[n_] := Block[{d=Divisors[n]}, DivisorSigma[1, n] - Plus@@Select[d, GCD[#, n/#] == 1&]]; Do[If[IntegerQ[Sqrt[usigma[n] && usigma[n] > 0]], AppendTo[lst, n]], {n, 2000}]; lst CROSSREFS Cf. A000203 (sigma:sum of divisors), A034448 (usigma:sum of unitary divisors). Cf. A064212 (sigma+usigma), A013929 (nonsquarefree numbers). Sequence in context: A141782 A093648 A171221 * A117101 A063840 A180117 Adjacent sequences:  A216256 A216257 A216258 * A216260 A216261 A216262 KEYWORD nonn AUTHOR Michel Lagneau, Mar 15 2013 STATUS approved

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Last modified November 26 09:38 EST 2020. Contains 338639 sequences. (Running on oeis4.)