A270308 -id:A270308 - cp's OEIS frontend

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

David A. Corneth and Amiram Eldar, Apr 20 2020

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?

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

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

Subsequence of A270308.
Cf. A074206, A122408, A163272.
Cf. A025487, A050324, A124010.

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
```

A340155 Numbers k such that A008480(k) > k.

Original entry on oeis.org

326918592000, 435891456000, 653837184000, 871782912000, 980755776000, 1089728640000, 1307674368000, 1401079680000, 1508855040000, 1525620096000, 1569209241600, 1587890304000, 1634592960000, 1710035712000, 1852538688000, 1868106240000, 1961511552000, 2070484416000

Offset: 1

Amiram Eldar, Dec 29 2020

nonn

a(1) is given in the paper by Montgomery and Tenenbaum (2017).
If k is a term then all the numbers below k with the same prime signature as k are also terms. In particular A046523(k) is a term.
1710035712000 is the least term which is not in A025487.
Are there any odd terms in this sequence? If there are, the least of them is a term of A147516 and has more than 90 prime divisors (counted with multiplicity).

			326918592000 is a term since A008480(326918592000) = 358500542400 > 326918592000.

Hugh L. Montgomery and Gérald Tenenbaum, On Multiplicative Compositions Of Integers, Mathematika, Vol. 63, No. 3 (2017), pp. 1081-1090; altermative link. See p. 1088.

Cf. A008480, A025487, A046523, A147516, A260987, A270308, A318170.

A385511 Numbers that are less than the number of their ordered factorizations into squarefree numbers greater than 1.

Original entry on oeis.org

2520, 5040, 7560, 10080, 10800, 12600, 15120, 20160, 21600, 22680, 23760, 25200, 27720, 30240, 32400, 35280, 37800, 43200, 45360, 47520, 50400, 52920, 55440, 60480, 64800, 65520, 70560, 71280, 75600, 79200, 83160, 86400, 88200, 90720, 95040, 98280, 100800, 105840

Offset: 1

Amiram Eldar, Jul 01 2025

nonn

Numbers k such that A050328(k) > k.
If k is a term then all the smaller numbers with the same prime signature (A118914) as k are also terms.
The least term that is not divisible by 5 is a(112) = 399168.
The least term that is not divisible by 3 is 144848704000.
The least odd term is A147516(43302) = 16639855392913235373515625.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A050328, A118914, A147516, A270308, A340155.

f[1] = 1; f[n_] := f[n] = DivisorSum[n, f[#] &, # < n && SquareFreeQ[n/#] &]; Select[Range[110000], f[#] > # &]

f(n) = if(n == 1, 1, sumdiv(n, d, if((d k;

cp's OEIS Frontend

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

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A340155 Numbers k such that A008480(k) > k.

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A385511 Numbers that are less than the number of their ordered factorizations into squarefree numbers greater than 1.

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Mathematica

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