A182292 Smallest odd number k such that is equal to the sum of its proper divisors greater than k^(1/n), or 0 if none exist.

34155, 407715, 8415

Offset

2,1

Comments

a(8) = 159030135. There is no n > 4 for which a(n) is smaller unless a(n) = 0. - Charles R Greathouse IV, Apr 25 2012

Other than a(2) to a(4) and a(8), there is no solution < 2*10^10 for a(n) up to a(1000). - Donovan Johnson, Aug 23 2012

From Alexander Violette, Feb 29 2024: (Start)

a(7) <= 7650499534755.

a(14) <= 221753170660847595. (End)

Links

Table of n, a(n) for n=2..4.

Example

The sum proper divisors of 407715 greater than 407715^(1/3) is 77 + 105 + 165 + 231 + 353 + 385 + 1059 + 1155 + 1765 + 2471 + 3883 + 5295 + 7413 + 11649 + 12355 + 19415 + 27181 + 37065 + 58245 + 81543 + 135905 = 407715.

Mathematica

t={}; d[n_]:= Select[Drop[Divisors[n], -1], #1>n^(1/p)&]; Do[s=Select[Range[1, 5*10^5, 2], #==Plus@@d[#]&];
  AppendTo[t, s], {p, 2, 4}]; Flatten[t]

Prog

(PARI) a(n)=my(t, k=8413); while(k+=2, if(sigma(k, -1)>2, if(ispower(k, n, &t), , t=k^(1/n)\1); if(sumdiv(k, d, if(d>t, d))==2*k, return(k)))) \\ Charles R Greathouse IV, Apr 25 2012

Crossrefs

See A182147 for more details for 34155.

Cf. A182311, A182286, A005231.

Sequence in context: A344666 A227699 A234820 * A206214 A227487 A204411

Adjacent sequences: A182289 A182290 A182291 * A182293 A182294 A182295

Keyword

nonn,bref

Author

Manuel Valdivia, Apr 24 2012

Status

approved