A383872 Nonprime numbers whose sum of proper divisors is a power of 4.

9, 12, 26, 56, 76, 122, 332, 992, 2042, 3344, 4336, 8186, 16256, 32762, 227744, 266176, 269072, 299576, 856544, 2097146, 5385812, 8388602, 16580864, 17895664, 19173944, 33554426, 61008020, 67100672, 201931760, 1074789376, 1108378592, 17179738112, 62472251540, 68700578816

Offset: 1

Author

Hans Ulrich Keller, May 13 2025

Keywords

• nonn

Comments

Includes 2*p for p in A135535. - Robert Israel, May 13 2025

From David A. Corneth, May 13 2025: (Start)

If a(n) = m*p where p is the largest prime divisor and has multiplicity 1 and s = sigma(m) then p = (4^k - s) / (s - m). Using this, a(44), a(45) and a(46) are at most 4971704751572, 44088037271892 and 44358570286896 respectively.

If a(n) is odd then a(n) is a perfect square. Proof: Suppose a(n) is not a perfect square. Then sigma(a(n)) is even and so sigma(a(n)) - a(n) = 4^k. As sigma(a(n)) - a(n) = 4^k then sigma(a(n)) - a(n) = 1. As a(n) is composite this has no solutions. (End)

For the first 43 terms, a(1) is the only square term and for the other terms, a(n) has a squarefree odd part. However, this is not always true as 44088037271892 (see above) is a term and its odd part is not squarefree. - Chai Wah Wu, May 19 2025

Examples

			12 is not prime; 12 has proper divisors 1, 2, 3, 4, and 6, with a sum of 16. This is a square number as well as a power of 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..43 (calculated from the b-file at A279731)

Crossrefs

Intersection of A048699 and A279731.

Cf. A001065, A135535.

Programs

Maple

filter:= proc(n) local s;
  s:= numtheory:-sigma(n)-n;
  s > 1 and s = 4^padic:-ordp(s,4)
end proc:
select(filter, [$4..10^7]); # Robert Israel, May 13 2025

Mathematica

Zweierpotenzen = {};
Quadratzahlen = {};
Beides = {};
For[k = 1, k <= 50000000, k++,
  SumET = Total[Divisors[k]] - k;
  If[IntegerQ[Log[2, SumET]] && PrimeQ[k] == False,
   AppendTo[Zweierpotenzen, k]];
  If[IntegerQ[Sqrt[SumET]] && PrimeQ[k] == False,
   AppendTo[Quadratzahlen, k ]]];
Beides = Intersection[Zweierpotenzen, Quadratzahlen];
Beides

PARI

isok(k) = if (!ispseudoprime(k), my(s=sigma(k)-k, z); issquare(s) && (ispower(s, , &z) && (z==2))); \\ Michel Marcus, May 13 2025

Extensions

a(27)-a(29) from Michel Marcus, May 13 2025

a(30)-a(34) from Amiram Eldar, May 13 2025