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

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..750 from Harry J. Smith)

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 *)

Prog

(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

Crossrefs

Cf. A034448, A048146.

Sequence in context: A178929 A145275 A234654 * A081635 A165614 A104875

Adjacent sequences: A064727 A064728 A064729 * A064731 A064732 A064733

Keyword

nonn

Author

Jason Earls, Oct 17 2001

Status

approved