cp's OEIS Frontend

A316225 Numbers k that divide the sum of sums of elements of all subsets of divisors of k (A229335).

%I A316225 #51 Jun 02 2020 08:00:51
%S A316225 1,2,4,6,8,16,24,28,32,40,64,96,120,128,224,256,288,360,384,496,512,
%T A316225 640,672,1024,1536,1792,1920,2016,2048,2176,3744,4096,4320,4680,5632,
%U A316225 5760,6144,6528,8128,8192,10240,10880,14336,15872,16384,16896,18432,18688
%N A316225 Numbers k that divide the sum of sums of elements of all subsets of divisors of k (A229335).
%C A316225 Harborth proved that this sequence is infinite. He showed that the terms are numbers n such that n|sigma(n)*2^(d(n) - 1), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203), and that the even terms, numbers of the form r*2^m where r is odd and m > 0, are those with m = ord_2(r/gcd(r, sigma(r)))*i with i = 1, 2, ... (ord_2(k) is the multiplicative order of 2 mod k, A002326). Thus this sequence includes all the powers of 2, all the numbers of the form n = 2^m*(2^(m + 1) - 1) which include the even perfect numbers.
%C A316225 Pollack and Pomerance call these numbers "H-perfect numbers". They prove that k is H-perfect if and only if denominator(sigma(k)/k) is a power of 2. - _Amiram Eldar_, Jun 02 2020
%H A316225 Amiram Eldar, <a href="/A316225/b316225.txt">Table of n, a(n) for n = 1..500</a>
%H A316225 Heiko Harborth, <a href="http://eudml.org/doc/141178">Eine Bemerkung zu den vollkommenen Zahlen</a>, Elemente der Mathematik, Vol. 31 (1976), pp. 119-121 (in German).
%H A316225 Paul Pollack and Carl Pomerance, <a href="http://www.emis.de/journals/INTEGERS/papers/a14self/a14self.Abstract.html">Prime-Perfect Numbers</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.
%t A316225 divSumSubQ[n_] := Divisible[DivisorSigma[1, n] * 2^(DivisorSigma[0, n] - 1), n]; Select[Range[100000], divSumSubQ]
%o A316225 (PARI) isok(n) = (sigma(n)*2^(numdiv(n)-1) % n) == 0; \\ _Michel Marcus_, Dec 21 2018
%Y A316225 Cf. A000005, A000203, A000396, A002326, A229335.
%K A316225 nonn
%O A316225 1,2
%A A316225 _Amiram Eldar_, Dec 21 2018