A064730 Numbers whose sum of nonunitary divisors and sum of unitary divisors are both positive squares.
15012, 124956, 128412, 135972, 186732, 219520, 241812, 377892, 414180, 420660, 447876, 453060, 453492, 497772, 504036, 515052, 523044, 528876, 544212, 658560, 776412, 826956, 1009792, 1020060, 1135836, 1191132, 1425060, 1467180, 1511892
Offset: 1
Keywords
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..750 from Harry J. Smith)
Programs
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Mathematica
sqQ[n_] := IntegerQ[Sqrt[n]]; f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e+1)-1)/(p-1); q[n_] := Module[{fct = FactorInteger[n], u}, If[AllTrue[fct[[;; , 2]], # == 1 &], False, u = Times @@ f1 @@@ fct; sqQ[u] && sqQ[Times @@ f2 @@@ fct - u]]]; Select[Range[10^6], q] (* Amiram Eldar, Jul 27 2024 *) -
PARI
{usigma(n, s=1, fac, i) = fac=factor(n); for(i=1,matsize(fac)[1],s=s*(1+fac[i,1]^fac[i,2])); return(s); } nu(n) = sigma(n)-usigma(n); for(n=1,10^8, if(nu(n)>0 && issquare(nu(n)) && issquare(usigma(n)), print1(n,","))) -
PARI
usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) } { n=0; for (m = 1, 10^9, u=usigma(m); nu=sigma(m) - u; if (nu>0 && issquare(nu) && issquare(u), write("b064730.txt", n++, " ", m); if (n==750, break)) ) } \\ Harry J. Smith, Sep 24 2009 -
PARI
is(n) = {my(f = factor(n), u); if(issquarefree(f), 0, u = prod(k=1, #f~, f[k, 1]^f[k, 2]+1); issquare(u) && issquare(sigma(f) - u));} \\ Amiram Eldar, Jul 27 2024