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%I #20 Mar 31 2021 02:58:03
%S 1,0,-3,-4,-10,-6,-14,-16,-18,-10,-22,-24,-39,-28,-15,-16,-34,-36,-57,
%T -60,-42,-22,-46,-48,-50,-26,-27,-28,-58,-90,-124,-128,-99,-68,-35,
%U -36,-74,-38,0,0,-41,-84,-129,-132
%N a(n) = n * A002321(n).
%C Equals row sums of triangle A143442.
%e First four terms = (1, 0, -3, -4) = (1*1, 2*0, 3*(-1), 4*(-1)), where the Mertens function A002321 = (1, 0, -1, -1, -2, -1, -2, -2, -2,...)
%e a(5) = -10 = sum of row 5 terms of triangle A143442: (5 - 5 - 5 + 0 - 5).
%t Table[n Plus @@ MoebiusMu[Range[n]], {n, 1, 80}] (* _Carl Najafi_, Aug 17 2011 *)
%o (PARI) a(n) = n*sum(k=1, n, moebius(k)); \\ _Michel Marcus_, Aug 22 2015
%o (Python)
%o from functools import lru_cache
%o @lru_cache(maxsize=None)
%o def A143443(n):
%o if n == 0:
%o return 0
%o c, j = n, 2
%o k1 = n//j
%o while k1 > 1:
%o j2 = n//k1 + 1
%o c += (j2-j)*A143443(k1)//k1
%o j, k1 = j2, n//j2
%o return n*(j-c) # _Chai Wah Wu_, Mar 30 2021
%Y Cf. A002321, A143442.
%K sign
%O 1,3
%A _Gary W. Adamson_, Aug 15 2008
%E More terms from _Carl Najafi_, Aug 17 2011
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