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

Author

Manuel Valdivia, Apr 24 2012

Keywords

• nonn
• bref

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)

Examples

			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.
		

Crossrefs

See A182147 for more details for 34155.

Cf. A182311, A182286, A005231.

Programs

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]

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