A062275 -id:A062275 - cp's OEIS frontend

A146748 Numbers of the form n^k * k^n, where n,k > 1.

16, 72, 256, 729, 800, 2304, 5184, 6272, 16384, 30375, 41472, 65536, 102400, 157464, 247808, 589824, 640000, 750141, 1384448, 3211264, 3359232, 5308416, 7372800, 9765625, 14348907, 16777216, 37879808, 39337984, 59049000, 84934656

Offset: 1

Howard Berman (howard_berman(AT)hotmail.com), Nov 01 2008

nonn

			2^2 * 2^2 = 16,
2^3 * 3^2 = 72.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007758, A062074, A062075, A062275.

N:= 10^20: # for terms <= N
S:= {}:
for n from 2 to ilog2(N) do
  for k from n do
    v:= n^k * k^n;
    if v > N then break fi;
    S:= S union {v};
od od:
sort(convert(S,list)); # Robert Israel, Oct 31 2023

A303990 Triangle, read by rows: n^k * k^n, for n >= 1 and k = 1..n.

Original entry on oeis.org

1, 2, 16, 3, 72, 729, 4, 256, 5184, 65536, 5, 800, 30375, 640000, 9765625, 6, 2304, 157464, 5308416, 121500000, 2176782336, 7, 6272, 750141, 39337984, 1313046875, 32934190464, 678223072849, 8, 16384, 3359232, 268435456, 12800000000, 440301256704, 12089663946752, 281474976710656

Offset: 1

Wolfdieter Lang, May 22 2018

Due to the symmetry of n^k * k^n under the exchange n <-> k, it is sufficient to consider n >= 1, and k = 1..n.
For the array n^k * k^n, with n >= 0 and k >= 0, read by antidiagonals, see the triangle A062275.
Thanks go to S. Heinemeyer for leading me to look at this.
The row sums give A303991.

			The triangle T(n, k) begins:
======================================================================
n\k |  1    2      3        4          5           6            7  ...
----+-----------------------------------------------------------------
1:  |  1
2:  |  2   16
3:  |  3   72    729
4:  |  4  256   5184    65536
5:  |  5  800  30375   640000    9765625
6:  |  6 2304 157464  5308416  121500000  2176782336
7:  |  7 6272 750141 39337984 1313046875 32934190464 678223072849
...
row n=8: 8, 16384, 3359232, 268435456, 12800000000, 440301256704, 12089663946752, 281474976710656;
row n=9: 9, 41472, 14348907, 1719926784, 115330078125, 5355700839936, 193010051319183, 5777633090469888, 150094635296999121;
row n=10: 10, 102400, 59049000, 10485760000, 976562500000, 60466176000000, 2824752490000000, 107374182400000000, 3486784401000000000, 100000000000000000000;
...

Cf. A062275, A303991.

/* As triangle */ [[n^k*k^n: k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, May 23 2018

Table[n^k k^n, {n, 10}, {k, n}] //Flatten (* Vincenzo Librandi, May 23 2018 *)

T(n, k) = n^k * k^n;
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 25 2018

T(n, k) = n^k * k^n, for n >= 1, k = 1..n.

A166974 Number of single-component graphs where the product of the valences of the nodes is n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 6, 1, 2, 2, 8, 1, 6, 1, 6, 2, 2, 1, 16, 2, 2, 4, 6, 1, 8, 1, 16, 2, 2, 2, 25, 1, 2, 2, 16, 1, 8, 1, 6, 6, 2, 1, 46, 2, 6, 2, 6, 1, 18, 2, 16, 2, 2, 1, 36, 1, 2, 6, 40, 2, 8, 1, 6, 2, 8, 1, 84, 1, 2, 6, 6, 2, 8, 1, 49, 12, 2, 1, 36, 2, 2, 2, 16, 1, 38, 2, 6, 2, 2, 2, 137

Offset: 0

Franklin T. Adams-Watters, Oct 26 2009

A single-component graph is any nonempty connected graph. If the empty graph was allowed, a(1) would be 2 instead of 1.
The sequence can be computed for n > 1 by looking at the graph that results when all valence 1 nodes are removed. This will be a connected graph, and labeling each node with its original valence, the product of the labels will be the original product. Each node must be labeled with at least its valence, and at least 2. Each such labeling, up to graph equivalence, uniquely defines the original graph, so we need only count the labelings for connected graphs with up to BigOmega(n) nodes.
Note, in particular, that a(n) = 1 for any prime, and 2 for any semiprime.
This product for the complete graph on n points is (n-1)^n. For the complete bipartite graph with n and m points in the parts the product is n^m*m^n. For the cyclic graph with n nodes it is 2^n.

Andrew Weimholt, Table of n, a(n) for n = 0..255

Cf. A000079, A001222 (BigOmega), A001349, A002905, A007778, A062275.

Corrected and extended by Andrew Weimholt, Oct 26 2009

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A146748 Numbers of the form n^k * k^n, where n,k > 1.

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A303990 Triangle, read by rows: n^k * k^n, for n >= 1 and k = 1..n.

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A166974 Number of single-component graphs where the product of the valences of the nodes is n.

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