A076407 Sum of perfect powers <= n.
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
Offset: 1
Keywords
Examples
Sum of the 8 perfect powers <= 32: a(32) = 1+4+8+9+16+25+27+32 = 122.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- Eric Weisstein's World of Mathematics, Perfect Powers.
Programs
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Maple
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: convert(V,list); # Robert Israel, Oct 19 2023 -
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
Formula
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