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Numbers having factorization Product_{i=1..m} p(i)^e(i) such that m > 1 and p(i) + e(i) is the same for each i.
2

%I #19 Jan 04 2021 18:20:39

%S 12,72,80,135,432,448,800,875,1701,2025,2160,2592,6272,8000,11264,

%T 15552,17303,30375,30625,35721,53248,56000,64800,80000,87808,93312,

%U 93347,108864,184877,212625,216513,247808,455625,559872,750141,756059,800000,859375

%N Numbers having factorization Product_{i=1..m} p(i)^e(i) such that m > 1 and p(i) + e(i) is the same for each i.

%H David A. Corneth, <a href="/A219302/b219302.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Donovan Johnson)

%e 12 = 2^2 * 3^1 and 2+2 = 3+1, so 12 is a term.

%e 64800 = 2^5 * 3^4 * 5^2 and 2+5 = 3+4 = 5+2, so 64800 is a term.

%e 212625 = 3^5 * 5^3 * 7^1 and 3+5 = 5+3 = 7+1, etc.

%e 750141 = 3^7 * 7^3 and 3+7 = 7+3, etc.

%e The first term with three prime factors is 2160 = 2^4 * 3^3 * 5^1; the first with four is 13608000 = 2^6 * 3^5 * 5^3 * 7^1.

%t fQ[n_] := Module[{f = FactorInteger[n]}, Length[f] > 1 && Length[Union[Plus @@@ f]] == 1]; Select[Range[2, 100000], fQ] (* _T. D. Noe_, Nov 21 2012 *)

%Y Cf. A219301 (includes primes and prime powers).

%K nonn

%O 1,1

%A _Carl R. White_, Nov 17 2012