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

%I A383872 #43 May 20 2025 00:27:51
%S A383872 9,12,26,56,76,122,332,992,2042,3344,4336,8186,16256,32762,227744,
%T A383872 266176,269072,299576,856544,2097146,5385812,8388602,16580864,
%U A383872 17895664,19173944,33554426,61008020,67100672,201931760,1074789376,1108378592,17179738112,62472251540,68700578816
%N A383872 Nonprime numbers whose sum of proper divisors is a power of 4.
%C A383872 Includes 2*p for p in A135535. - _Robert Israel_, May 13 2025
%C A383872 From _David A. Corneth_, May 13 2025: (Start)
%C A383872 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.
%C A383872 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)
%C A383872 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
%H A383872 Amiram Eldar, <a href="/A383872/b383872.txt">Table of n, a(n) for n = 1..43</a> (calculated from the b-file at A279731)
%e A383872 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.
%p A383872 filter:= proc(n) local s;
%p A383872   s:= numtheory:-sigma(n)-n;
%p A383872   s > 1 and s = 4^padic:-ordp(s,4)
%p A383872 end proc:
%p A383872 select(filter, [$4..10^7]); # _Robert Israel_, May 13 2025
%t A383872 Zweierpotenzen = {};
%t A383872 Quadratzahlen = {};
%t A383872 Beides = {};
%t A383872 For[k = 1, k <= 50000000, k++,
%t A383872   SumET = Total[Divisors[k]] - k;
%t A383872   If[IntegerQ[Log[2, SumET]] && PrimeQ[k] == False,
%t A383872    AppendTo[Zweierpotenzen, k]];
%t A383872   If[IntegerQ[Sqrt[SumET]] && PrimeQ[k] == False,
%t A383872    AppendTo[Quadratzahlen, k ]]];
%t A383872 Beides = Intersection[Zweierpotenzen, Quadratzahlen];
%t A383872 Beides
%o A383872 (PARI) isok(k) = if (!ispseudoprime(k), my(s=sigma(k)-k, z); issquare(s) && (ispower(s, , &z) && (z==2))); \\ _Michel Marcus_, May 13 2025
%Y A383872 Intersection of A048699 and A279731.
%Y A383872 Cf. A001065, A135535.
%K A383872 nonn
%O A383872 1,1
%A A383872 _Hans Ulrich Keller_, May 13 2025
%E A383872 a(27)-a(29) from _Michel Marcus_, May 13 2025
%E A383872 a(30)-a(34) from _Amiram Eldar_, May 13 2025