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A063936 Numbers k such that the sum of unitary proper divisors of k is a square > 1.

%I A063936 #24 Mar 14 2020 05:09:05
%S A063936 15,26,44,56,95,96,119,122,124,140,143,194,215,216,236,287,304,364,
%T A063936 386,407,495,511,527,551,556,560,575,639,740,752,764,780,791,794,815,
%U A063936 871,900,935,936,992,1004,1036,1116,1159,1196,1199,1232,1295,1328,1346
%N A063936 Numbers k such that the sum of unitary proper divisors of k is a square > 1.
%C A063936 A unitary divisor of n is a divisor d of n such that gcd(d, n/d) = 1.
%H A063936 Amiram Eldar, <a href="/A063936/b063936.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%F A063936 A034460(a(n)) > 1 and A010052(A034460(a(n))) = 1. - _Reinhard Zumkeller_, Aug 15 2012
%e A063936 The unitary divisors of 15 are 1,3,5,15 and then the unitary aliquot part is 9 which is a square.
%t A063936 us[1] = 0; us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; Select[Range[1500], (s = us[#]) > 1 && IntegerQ@Sqrt[s] &] (* _Amiram Eldar_, Mar 14 2020 *)
%o A063936 (PARI) us(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d));
%o A063936 j=[]; for(n=1,3000, if(us(n)-n > 1 && issquare(us(n)-n),j=concat(j,n))); j
%o A063936 (PARI) us(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d))
%o A063936 { n=0; for (m=1, 10^9, u=us(m) - m; if (issquare(u) && u > 1, write("b063936.txt", n++, " ", m); if (n==1000, break)) ) } \\ _Harry J. Smith_, Sep 03 2009
%o A063936 (Haskell)
%o A063936 import Data.List (findIndices)
%o A063936 a063936 n = a063936_list !! (n-1)
%o A063936 a063936_list = map (+ 1) $
%o A063936                findIndices (\x -> x > 1 && a010052 x == 1) a034460_list
%o A063936 -- _Reinhard Zumkeller_, Aug 15 2012
%Y A063936 Cf. A034448, A063937.
%K A063936 easy,nonn
%O A063936 1,1
%A A063936 _Felice Russo_, Aug 31 2001
%E A063936 More terms from _Jason Earls_, Sep 04 2001