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A076407
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Sum of perfect powers <= n.
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2
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1, 1, 1, 5, 5, 5, 5, 13, 22, 22, 22, 22, 22, 22, 22, 38, 38, 38, 38, 38, 38, 38, 38, 38, 63, 63, 90, 90, 90, 90, 90, 122, 122, 122, 122, 158, 158, 158, 158, 158, 158, 158, 158, 158, 158, 158, 158, 158, 207, 207, 207, 207, 207, 207, 207, 207, 207, 207, 207, 207, 207
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) = 1 - Sum_{k=2..floor(log_2(n))} mu(k) * (F(k, floor(n^(1/k))) - 1), where F(k, n) = Sum_{j=1..n} j^k = (Bernoulli(k+1, n+1) - Bernoulli(k+1, 1))/(k+1). - Daniel Suteu, Aug 19 2023
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EXAMPLE
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Sum of the 8 perfect powers <= 32: a(32) = 1+4+8+9+16+25+27+32 = 122.
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MAPLE
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N:= 100: # for a(1)..a(N)
V:= Vector(N, 1):
pps:= {seq(seq(x^k, k=2..floor(log[x](N))), x=2..floor(sqrt(N)))}:
for y in pps do
V[y..N]:= V[y..N] +~ y
od:
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PROG
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(PARI)
F(k, n) = (subst(bernpol(k+1), x, n+1) - subst(bernpol(k+1), x, 1)) / (k+1);
a(n) = 1 - sum(k=2, logint(n, 2), moebius(k) * (F(k, sqrtnint(n, k)) - 1)); \\ Daniel Suteu, Aug 19 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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