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A089361
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Numbers of pairs (i, j), i, j > 1, such that i^j <= n.
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4
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0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15
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OFFSET
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1,8
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COMMENTS
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These numbers are related to the divergent series r sum(n^(1/k)) = n^(1/2) + n^(1/3) + ... + n^(1/r) for abs(n) > 0 and r=sqrt(n). Notice some numbers are missing, such as 4, 11, 12, 14.
Gaps (i.e., a(n) - a(n-1) > 1) occur for values of n > 1 in A117453. a(n) - a(n-1) = number of factors of j > 1, for the j in the pair (i,j) with the smallest value of i. Where n = A117453(x), a(n) = a(n-1) + A175066(x). For example: n = 64, a(64) = 13, a(63) = 10, 13 - 10 = 3; 64 = 2^6, 6 has three factors (2,3,6), corresponding to the three perfect powers for 64 (2^6, 4^3, 8^2). Also, A117453(3) = 64 and A175066(3) = 3. - Doug Bell, Jun 23 2015
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LINKS
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FORMULA
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a(1) = 0; for n > 1, if n not in A001597, a(n) = a(n-1), otherwise a(n) = a(n-1) + number of factors of j > 1 (A000005(j) - 1), for the j in the positive integer pair (i,j) where i^j = n with the smallest value of i. - Doug Bell, Jun 23 2015
a(n) = Sum(j = 2 .. floor(log_2(n)), floor(n^(1/j) - 1)). - Robert Israel, Jun 24 2015
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EXAMPLE
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There are 5 perfect powers greater than 1 that are less than or equal to 16: 2^2, 2^3, 2^4, 3^2, 4^2, ergo the first 5 in the table.
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MAPLE
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N:= 1000; # to get a(1) to a(N)
B:= Vector(N);
for i from 2 to floor(sqrt(N)) do
for j from 2 while i^j <= N do
B[i^j]:= B[i^j]+1
od
od:
convert(map(round, Statistics:-CumulativeSum(B)), list); # Robert Israel, Jun 24 2015
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PROG
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(PARI) plessn(n, m) = { for(k=1, n, s=0; r = floor(sqrt(k)); for(x=m, r, for(y=2, r, p = floor(x^y); if(p<=k, s++) ) ); print1(s", ") ) }
(PARI) A = vector(100); for (p = 2, 6, i = 2; while (i^p <= 100, A[i^p]++; i++)); for (n = 2, 100, A[n] += A[n - 1]); \\ David Wasserman
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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