A072652
Ordered solutions greater than 1 to b^c=c^d, with multiplicity.
Original entry on oeis.org
4, 16, 16, 27, 64, 256, 256, 729, 1024, 3125, 4096, 4096, 16384, 19683, 46656, 65536, 65536, 65536, 262144, 531441, 823543, 1048576, 1048576, 4194304, 9765625, 14348907, 16777216, 16777216, 16777216, 67108864, 268435456, 268435456
Offset: 2
65536 appears three times because 256^2=2^16, 2^16=16^4 and 16^4=4^8 are all solutions, each coming to 65536.
A072653
Uniqued integer solutions n to n = b^c = c^d.
Original entry on oeis.org
1, 4, 16, 27, 64, 256, 729, 1024, 3125, 4096, 16384, 19683, 46656, 65536, 262144, 531441, 823543, 1048576, 4194304, 9765625, 14348907, 16777216, 67108864, 268435456, 387420489, 1073741824, 2176782336, 4294967296, 10000000000
Offset: 1
1 is included because of solutions of the form b^0 = 0^0, 1^c = c^0 and 1^1 = 1^d; 4 since 2^2 = 2^2; 16 since 2^4 = 4^2 and 4^2 = 2^4; 27 since 3^3 = 3^3; 64 since 8^2 = 2^6; etc.
The 10th element is n = 4096 with i = 12 and j = 6 because (4096^12)^(4096^(1/12)) = (4096^6)^(4096^(1/6)).
A111260
Numbers of the form (m^n)/(n^m) with m > 0 and n>1.
Original entry on oeis.org
1, 4, 16, 27, 256, 729, 65536, 2985984, 4194304, 9765625, 134217728, 387420489, 2176782336, 24794911296, 30517578125, 104857600000, 678223072849, 2641807540224, 7625597484987, 17592186044416, 281474976710656
Offset: 1
The 4th element is a=27 with m = 3 and n = 9 because 3^9/9^3 = 27.
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a:=proc(N) local a, m,n; for m from 1 to N do for n from 2 to N do a:=(m^n)/(n^m); if(floor(a)=a)then print(a) fi; od; od; end: # convert into set # sort set
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Take[ Select[ Union@Flatten@Table[(m^n)/(n^m), {m, 35}, {n, 2, 35}], IntegerQ[ # ] &], 21] (* Robert G. Wilson v, Nov 17 2005 *)
Showing 1-3 of 3 results.
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