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

15, 26, 44, 56, 95, 96, 119, 122, 124, 140, 143, 194, 215, 216, 236, 287, 304, 364, 386, 407, 495, 511, 527, 551, 556, 560, 575, 639, 740, 752, 764, 780, 791, 794, 815, 871, 900, 935, 936, 992, 1004, 1036, 1116, 1159, 1196, 1199, 1232, 1295, 1328, 1346

Offset: 1

Author

Felice Russo, Aug 31 2001

Keywords

• easy
• nonn

Comments

A unitary divisor of n is a divisor d of n such that gcd(d, n/d) = 1.

Examples

			The unitary divisors of 15 are 1,3,5,15 and then the unitary aliquot part is 9 which is a square.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Crossrefs

Cf. A034448, A063937.

Programs

Haskell

import Data.List (findIndices)
a063936 n = a063936_list !! (n-1)
a063936_list = map (+ 1) $
               findIndices (\x -> x > 1 && a010052 x == 1) a034460_list
-- Reinhard Zumkeller, Aug 15 2012

Mathematica

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 *)

PARI

us(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d));
j=[]; for(n=1,3000, if(us(n)-n > 1 && issquare(us(n)-n),j=concat(j,n))); j

PARI

us(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d))
{ 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

Formula

A034460(a(n)) > 1 and A010052(A034460(a(n))) = 1. - Reinhard Zumkeller, Aug 15 2012

Extensions

More terms from Jason Earls, Sep 04 2001