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%I #6 May 08 2020 04:42:43
%S 18,36,75,100,200,224,225,441,560,1183,1344,1920,3025,8281,26011,
%T 34606,64009,72030,76895,115351,197173,280041,494209,538265,1168561,
%U 1947271,2927521,3575881,3613153,3780295,4492125,7295401,10665331,11580409,12511291,13476375,15381133
%N Composite numbers m such that tau_k(m) = m for some k, where tau_k is the k-th Piltz divisor function (A077592).
%C The prime numbers are excluded from this sequence since tau_p(p) = p for all primes p.
%C The corresponding values of k are 3, 3, 5, 4, 4, 4, 5, 6, 4, 13, 4, 4, 10, 13, 37, 11, 22, 7, 13, 61, 73, 17, 37, 13, 46, 157, 58, 61, 193, 29, 9, 73, 277, 82, 37, 9, 313, ...
%e 18 is in the sequence since tau_3(18) = A007425(18) = 18.
%t fun[e_, k_] := Times @@ (Binomial[# + k - 1, k - 1] & /@ e); tau[n_, k_] := fun[ FactorInteger[n][[;; , 2]], k]; aQ[n_] := CompositeQ[n] && Module[{k = 2}, While[(t = tau[n, k]) < n, k++]; t == n]; Select[Range[10^5], aQ]
%Y Cf. A097989.
%Y Cf. A007425, A007426, A034695, A061200, A077592, A111217, A111218, A111219, A111220, A111221, A111306, A163272.
%K nonn
%O 1,1
%A _Amiram Eldar_, Sep 25 2019
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