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%I #34 Aug 18 2015 12:31:48
%S 1,0,0,1,0,0,0,1,1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,
%T 0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0
%N a(1) = 1, for n > 1: a(n) is the number of ways to write n as a nontrivial perfect power.
%C a(n) = number of integer pairs (i,j) for distinct values of i where i > 0, j > 1 and n = i^j. Since 1 = 1^r for all real values of r, the requirement for a distinct i causes a(1) = 1 instead of a(1) = infinity.
%C Alternatively, the sequence can be defined as: a(1) = 1, for n > 1: a(n) = number of pairs (i,j) such that i > 0, j > 1 and n = i^j.
%C A007916 = n, where a(n) = 0.
%C A001597 = n, where a(n) > 0.
%C A175082 = n, where n = 1 or a(n) = 0.
%C A117453 = n, where n = 1 or a(n) > 1.
%C A175065 = n, where n > 1 and a(n) > 0 and this is the first occurrence in this sequence of a(n).
%C A072103 = n repeated a(n) times where n > 1.
%C A075802 = min(1, a(n)).
%C A175066 = a(n), where n = 1 or a(n) > 1. This sequence is an expansion of A175066.
%C A253642 = 0 followed by a(n), where n > 1 and a(n) > 0.
%C A175064 = a(1) followed by a(n) + 1, where n > 1 and a(n) > 0.
%C Where n > 1, A001597(x) = n (which implies a(n) > 0), i = A025478(x) and j = A253641(n), then a(n) = A000005(j) - 1, which is the number of factors of j greater than 1. The integer pair (i,j) comprises the smallest value i and the largest value j where i > 0, j > 1 and n = i^j. The a(n) pairs of (a,b) where a > 0, b > 1 and n = a^b are formed with b = each of the a(n) factors of j greater than 1. Examples for n = {8,4096}:
%C a(8) = 1, A001597(3) = 8, A025478(3) = 2, A253641(8) = 3, 8 = 2^3 and A000005(3) - 1 = 1 because there is one factor of 3 greater than 1 [3]. The set of pairs (a,b) is {(2,3)}.
%C a(4096) = 5, A001597(82) = 4096, A025478(82) = 2, A253641(4096) = 12, 4096 = 2^12 and A000005(12) - 1 = 5 because there are five factors of 12 greater than 1 [2,3,4,6,12]. The set of pairs (a,b) is {(64,2),(16,3),(8,4),(4,6),(2,12)}.
%C A023055 = the ordered list of x+1 with duplicates removed, where x is the number of consecutive zeros appearing in this sequence between any two nonzero terms.
%C A070428(x) = number of terms a(n) > 0 where n <= 10^x.
%C a(n) <= A188585(n).
%H Doug Bell, <a href="/A259362/b259362.txt">Table of n, a(n) for n = 1..5000</a>
%F a(1) = 1, for n > 1: a(n) = A000005(A253641(n)) - 1.
%F If n not in A001597, then a(n) = 0, otherwise a(n) = A175064(x) - 1 where A001597(x) = n.
%e a(6) = 0 because there is no way to write 6 as a nontrivial perfect power.
%e a(9) = 1 because there is one way to write 9 as a nontrivial perfect power: 3^2.
%e a(16) = 2 because there are two ways to write 16 as a nontrivial perfect power: 2^4, 4^2.
%Y Cf. A001597, A007916, A023055, A025478, A070428, A072103, A075090, A075109, A075802, A117453, A175064, A175066, A175082, A188585, A253641, A253642.
%K nonn
%O 1,16
%A _Doug Bell_, Jun 24 2015
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