A334256 Numbers k such that H(k) = 2*k, where H(k) is the number of ordered factorizations of k (A074206).

3072, 1310720, 469762048, 48378511622144, 14636698788954112, 1115414963960152064, 1254378597012249509888, 358899852698093036240896, 28472620903563746322679857152

Offset: 1

Author

David A. Corneth and Amiram Eldar, Apr 20 2020

Keywords

• nonn
• more

Comments

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

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.

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?

Examples

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

Links

• David A. Corneth, List of found terms in form of oddpart * 2^(multiplicity of 2)

Crossrefs

Subsequence of A270308.

Cf. A074206, A122408, A163272.

Cf. A025487, A050324, A124010.

Programs

Mathematica

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

PARI

is(n) = A074206(n) == n<<1