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a(n) is the number of values of m such that the sum of proper bi-unitary divisors of m (A331970) is n.
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%I #9 Feb 04 2020 03:57:04

%S 0,0,0,0,1,1,2,1,2,2,2,2,2,2,2,3,1,2,1,3,2,3,1,3,1,3,2,4,1,6,1,4,2,4,

%T 2,5,0,3,1,4,2,5,1,4,2,4,1,6,2,5,2,5,2,8,1,6,1,4,2,7,1,5,3,5,2,8,0,5,

%U 1,6,1,8,2,5,3,6,3,9,0,6,2,5,1,9,1,7,1

%N a(n) is the number of values of m such that the sum of proper bi-unitary divisors of m (A331970) is n.

%C The bi-unitary version of A048138.

%C The offset is 2 as in A048138 since there are infinitely many numbers k (the primes and squares of primes) for which A331970(k) = 1.

%H Amiram Eldar, <a href="/A331971/b331971.txt">Table of n, a(n) for n = 2..30000</a>

%e a(8) = 2 since 8 is the sum of the proper bi-unitary divisors of 2 numbers: 10 (1 + 2 + 5) and 12 (1 + 3 + 4).

%t fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); bs[n_] := bsigma[n] - n; m = 300; v = Table[0, {m}]; Do[b = bs[k]; If[2 <= b <= m, v[[b]]++], {k, 1, m^2}]; Rest @ v

%Y Cf. A048138, A188999, A222266, A324938, A331970, A331973.

%K nonn

%O 2,7

%A _Amiram Eldar_, Feb 03 2020