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Numbers k such that H(k) = 2*k, where H(k) is the number of ordered factorizations of k (A074206).
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%I #10 Apr 21 2020 13:52:32

%S 3072,1310720,469762048,48378511622144,14636698788954112,

%T 1115414963960152064,1254378597012249509888,358899852698093036240896,

%U 28472620903563746322679857152

%N Numbers k such that H(k) = 2*k, where H(k) is the number of ordered factorizations of k (A074206).

%C If p is an odd prime then 2^(4*p - 2) * p is a term, hence this sequence is infinite.

%C Since A074206(k) depends only on the prime signature (A124010) of k, then each term is of the form A050324(k)/2 = A074206(A025487(k))/2.

%C Besides terms of the form 2^(4*p - 2) * p at least 79 terms not of this form are known. For example, 1115414963960152064 = 2^46 * 11^2 * 131 is a term not of this form. To ease the search, can we narrow the possible prime signatures of terms?

%H David A. Corneth, <a href="/A334256/a334256.gp.txt">List of found terms in form of oddpart * 2^(multiplicity of 2)</a>

%e 3072 is a term since A074206(3072) = 6144 = 2 * 3072.

%t h[1] = 1; h[n_] := h[n] = DivisorSum[n, h[#] &, # < n &]; Select[Range[1.5*10^6], h[#] == 2*# &]

%o (PARI) is(n) = A074206(n) == n<<1

%Y Subsequence of A270308.

%Y Cf. A074206, A122408, A163272.

%Y Cf. A025487, A050324, A124010.

%K nonn,more

%O 1,1

%A _David A. Corneth_ and _Amiram Eldar_, Apr 20 2020