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A326988 Sum of nonpowers of 2 dividing n.

%I A326988 #63 Sep 08 2022 08:46:24
%S A326988 0,0,3,0,5,9,7,0,12,15,11,21,13,21,23,0,17,36,19,35,31,33,23,45,30,39,
%T A326988 39,49,29,69,31,0,47,51,47,84,37,57,55,75,41,93,43,77,77,69,47,93,56,
%U A326988 90,71,91,53,117,71,105,79,87,59,161,61,93,103,0,83,141,67,119,95,141,71,180,73,111,123,133,95,165,79,155
%N A326988 Sum of nonpowers of 2 dividing n.
%C A326988 In other words: a(n) is the sum of the divisors of n that are not powers of 2.
%C A326988 a(n) is also the sum of odd divisors greater than 1 of n, multiplied by the sum of the divisors of n that are powers of 2.
%C A326988 a(n) = 0 if and only if n is a power of 2.
%C A326988 a(n) = n if and only if n is an odd prime.
%C A326988 From _Bernard Schott_, Sep 17 2019: (Start)
%C A326988 a(n) = 3*n/2 if and only if n is an even semiprime greater than or equal to 6 (A100484).
%C A326988 a(n) = n + sqrt(n) if and only if n is the square of an odd prime (see A001248 without its first term). (End)
%H A326988 Robert Israel, <a href="/A326988/b326988.txt">Table of n, a(n) for n = 1..10000</a>
%F A326988 a(n) = A000203(n) - A038712(n).
%F A326988 a(n) = (A000593(n) - 1)*A038712(n).
%F A326988 a(n) = A326990(n)*A038712(n).
%F A326988 a(n) = Sum_{d|n, d > 1} d * (1 - [rad(d) = 2]), where rad is the squarefree kernel (A007947) and [] is the Iverson bracket, which gives 1 if the condition is true, 0 if it's false. - _Wesley Ivan Hurt_, Apr 29 2020
%e A326988 For n = 18 the divisors of 18 are [1, 2, 3, 6, 9, 18]. There are four divisors of 18 that are not powers of 2, they are [3, 6, 9, 18]. The sum of them is 3 + 6 + 9 + 18 = 36, so a(18) = 36.
%e A326988 On the other hand, the sum of odd divisors greater than 1 of 18 is 3 + 9 = 12, and the sum of the divisors of 18 that are powers of 2 is 1 + 2 = 3, then we have that 12 * 3 = 36, so a(18) = 36.
%p A326988 f:= n -> numtheory:-sigma(n) - 2^(1+padic:-ordp(n,2))+1:
%p A326988 map(f, [$1..100]); # _Robert Israel_, Apr 29 2020
%t A326988 Table[DivisorSigma[1, n] - Denominator[DivisorSigma[1, 2n]/DivisorSigma[1, n]], {n, 100}] (* _Wesley Ivan Hurt_, Aug 24 2019 *)
%o A326988 (Magma) sol:=[];  m:=1;  for n in [1..80] do v:=Set(Divisors(n)) diff {2^k:k in [0..Floor(Log(2,n))]};  sol[m]:=&+v; m:=m+1; end for; sol; // _Marius A. Burtea_, Aug 24 2019
%o A326988 (PARI) ispp2(n) = (n==1) || (isprimepower(n, &p) && (p==2));
%o A326988 a(n) = sumdiv(n, d, if (!ispp2(d), d)); \\ _Michel Marcus_, Aug 26 2019
%o A326988 (Scala) def divisors(n: Int): IndexedSeq[Int] = (1 to n).filter(n % _ == 0)
%o A326988 (1 to 80).map(divisors(_).filter(n => n != Integer.highestOneBit(n)).sum) // _Alonso del Arte_, Apr 29 2020
%o A326988 (Python)
%o A326988 from sympy import divisor_sigma
%o A326988 def A326988(n): return divisor_sigma(n)-(n^(n-1)) # _Chai Wah Wu_, Aug 04 2022
%Y A326988 Row sums of A326989.
%Y A326988 Cf. A000079, A000203, A000593, A038712, A057716, A065091, A326987 (number), A326990.
%K A326988 nonn,easy
%O A326988 1,3
%A A326988 _Omar E. Pol_, Aug 18 2019