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Numbers m such that A327566(m) = Sum_{k=1..m} isigma(k) is divisible by m, where isigma(k) is the sum of infinitary divisors of k (A049417).
1

%I #16 Sep 20 2019 08:58:48

%S 1,2,160,285,2340,2614,8903,81231,171710,182712,434887,2651907,

%T 56517068,143714354,922484770,5162883263,39421525873

%N Numbers m such that A327566(m) = Sum_{k=1..m} isigma(k) is divisible by m, where isigma(k) is the sum of infinitary divisors of k (A049417).

%C The infinitary version of A056550.

%C The corresponding quotients, A327566(a(n))/a(n), are 1, 2, 118, 209, 1711, 1910, 6506, 59357, 125473, 133513, 317781, 1937798, 41298052, 105014703, 674076450, 3772612983, 28806028088, ...

%e 2 is in the sequence since isigma(1) + isigma(2) = 1 + 3 = 4 is divisible by 2.

%t f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], _?(# == 1 &)])); isigma[1] = 1; isigma[n_] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); seq = {}; s = 0; Do[s = s + isigma [n]; If[Divisible[s, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

%Y Cf. A049417 (isigma), A327566 (sums of isigma).

%Y Cf. A056550 (corresponding with sigma), A064611 (unitary), A307043 (exponential), A307161 (bi-unitary).

%K nonn,more

%O 1,2

%A _Amiram Eldar_, Sep 20 2019