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Expansion of 1/(1 - Sum_{k>=2} (1 - floor(2/d(k)))*x^k), where d(k) is the number of divisors (A000005).
1

%I #8 Feb 16 2025 08:33:39

%S 1,0,0,0,1,0,1,0,2,1,3,0,5,2,8,3,13,5,22,10,34,18,58,31,94,57,153,99,

%T 254,172,417,302,685,523,1136,901,1872,1557,3097,2673,5133,4577,8505,

%U 7843,14109,13380,23440,22816,38953,38855,64789,66053,107871,112190,179664,190369,299478,322683,499501,546548

%N Expansion of 1/(1 - Sum_{k>=2} (1 - floor(2/d(k)))*x^k), where d(k) is the number of divisors (A000005).

%C Number of compositions (ordered partitions) of n into composite parts (A002808).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompositeNumber.html">Composite Number</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F G.f.: 1/(1 - Sum_{k>=2} (1 - floor(2/d(k)))*x^k).

%e a(10) = 3 because we have [10], [6, 4] and [4, 6].

%t nmax = 59; CoefficientList[Series[1/(1 - Sum[(1 - Floor[2/DivisorSigma[0, k]]) x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

%o (PARI) x='x+O('x^60); Vec(1/(1 - sum(k=2, 59, (1 - 2\numdiv(k))*x^k))) \\ _Indranil Ghosh_, Apr 03 2017

%Y Cf. A000005, A002808, A023360, A023895, A052284.

%K nonn,changed

%O 0,9

%A _Ilya Gutkovskiy_, Jan 05 2017