A327774 Composite numbers m such that tau_k(m) = m for some k, where tau_k is the k-th Piltz divisor function (A077592).

18, 36, 75, 100, 200, 224, 225, 441, 560, 1183, 1344, 1920, 3025, 8281, 26011, 34606, 64009, 72030, 76895, 115351, 197173, 280041, 494209, 538265, 1168561, 1947271, 2927521, 3575881, 3613153, 3780295, 4492125, 7295401, 10665331, 11580409, 12511291, 13476375, 15381133

Offset: 1

Author

Amiram Eldar, Sep 25 2019

Keywords

• nonn

Comments

The prime numbers are excluded from this sequence since tau_p(p) = p for all primes p.

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, ...

Examples

			18 is in the sequence since tau_3(18) = A007425(18) = 18.
		

Crossrefs

Cf. A097989.

Cf. A007425, A007426, A034695, A061200, A077592, A111217, A111218, A111219, A111220, A111221, A111306, A163272.

Programs

Mathematica

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]