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A085410
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Total number of parts in all partitions of n into relatively prime parts.
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4
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1, 2, 5, 9, 19, 27, 53, 74, 122, 170, 274, 355, 555, 724, 1043, 1377, 1964, 2487, 3497, 4429, 5993, 7622, 10205, 12701, 16831, 20964, 27166, 33756, 43452, 53296, 68134, 83464, 105086, 128495, 160803, 195006, 242811, 293701, 362026, 436842, 536103
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OFFSET
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1,2
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LINKS
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FORMULA
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EXAMPLE
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Partitions of 6 into relatively prime parts are: 1+1+1+1+1+1, 1+1+1+1+2, 1+1+2+2, 1+1+1+3, 1+2+3, 1+1+4, 1+5; total number of parts is a(6)=6+5+4+4+3+3+2=27.
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MATHEMATICA
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f[n_] := Sum[DivisorSigma[0, m] PartitionsP[n - m], {m, 1, n}]; MT[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu /@ (n/d)*f /@ d)]; Table[ MT[n], {n, 1, 41}]
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PROG
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(PARI) a006128(n) = sum(m=1, n, numdiv(m)*numbpart(n - m));
a(n) = sumdiv(n, d, moebius(n/d)*a006128(d)); \\ Indranil Ghosh, Apr 25 2017
(Python)
from sympy import divisors, divisor_count, npartitions, mobius
def a006128(n): return sum([divisor_count(m)*npartitions(n - m) for m in range(1, n + 1)])
def a(n): return sum([mobius(n/d)*a006128(d) for d in divisors(n)]) # Indranil Ghosh, Apr 25 2017
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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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EXTENSIONS
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STATUS
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approved
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