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Numbers having factorization Product_{i=1..m} p(i)^e(i) such that p(i) + e(i) is the same for each i.
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%I #18 Jan 03 2021 14:08:53

%S 1,2,3,4,5,7,8,9,11,12,13,16,17,19,23,25,27,29,31,32,37,41,43,47,49,

%T 53,59,61,64,67,71,72,73,79,80,81,83,89,97,101,103,107,109,113,121,

%U 125,127,128,131,135,137,139,149,151,157,163,167,169,173,179,181

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

%C All primes and powers of primes are necessarily in this sequence.

%H T. D. Noe, <a href="/A219301/b219301.txt">Table of n, a(n) for n = 1..1000</a>

%e 80 = 2^4 * 5^1 and 2+4 = 5+1, so 80 is a term.

%e 101 = 101^1 and 101 + 1 = no other sums due to being prime, so 101 is a term.

%e 128 = 2^7 and 2+7 = no other sums due to being a prime power, so 128 is a term.

%e 135 = 3^3 * 5^1 and 3+3 = 5+1, so 135 is a term.

%e The first term with three unique prime factors is 2160 = 2^4 * 3^3 * 5^1, since 2+4 = 3+3 = 5+1.

%t Join[{1}, Select[Range[2, 200], Length[Union[Plus @@@ FactorInteger[#]]] == 1 &] (* _T. D. Noe_, Nov 21 2012 *)

%Y Cf. A219302 (subset excludes primes and prime powers).

%K nonn

%O 1,2

%A _Carl R. White_, Nov 17 2012