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%I #35 Aug 07 2022 16:23:22
%S 1,4,6,11,10,20,14,26,23,32,22,50,26,44,44,57,34,72,38,78,60,68,46,
%T 112,59,80,76,106,58,136,62,120,92,104,92,173,74,116,108,172,82,184,
%U 86,162,150,140,94,238,111,180,140,190,106,232,140,232,156,176,118,324,122,188
%N a(n) = 2*sigma(n) - tau(n), where tau(n) is the number of divisors of n (A000005) and sigma(n) is the sum of divisors of n (A000203).
%C Row sums of A129234. - _Emeric Deutsch_, Apr 17 2007
%C Equals row sums of A130307. - _Gary W. Adamson_, May 20 2007
%C Equals row sums of triangle A143315. - _Gary W. Adamson_, Aug 06 2008
%C Equals A051731 * (1, 3, 5, 7, ...); i.e., the inverse Mobius transform of the odd numbers. Example: a(4) = 11 = (1, 1, 0, 1) * (1, 3, 5, 7) = (1 + 3 + 0 + 7), where (1, 1, 0, 1) = row 4 of A051731. - _Gary W. Adamson_, Aug 17 2008
%C Equals row sums of triangle A143594. - _Gary W. Adamson_, Aug 26 2008
%H G. C. Greubel, <a href="/A129235/b129235.txt">Table of n, a(n) for n = 1..5000</a>
%F G.f.: Sum_{k>=1} z^k*(k-(k-1)*z^k)/(1-z^k)^2. - _Emeric Deutsch_, Apr 17 2007
%F G.f.: Sum_{n>=1} x^n*(1+x^n)/(1-x^n)^2. - _Joerg Arndt_, May 25 2011
%F L.g.f.: -log(Product_{k>=1} (1 - x^k)^(2-1/k)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, Mar 18 2018
%F a(n) = A222548(n) - A222548(n-1). - _Ridouane Oudra_, Jul 11 2020
%e a(4) = 2*sigma(4) - tau(4) = 2*7 - 3 = 11.
%p with(numtheory): seq(2*sigma(n)-tau(n),n=1..75); # _Emeric Deutsch_, Apr 17 2007
%p G:=sum(z^k*(k-(k-1)*z^k)/(1-z^k)^2,k=1..100): Gser:=series(G,z=0,80): seq(coeff(Gser,z,n),n=1..75); # _Emeric Deutsch_, Apr 17 2007
%t a[n_] := DivisorSum[2n, If[EvenQ[#], #-1, 0]&]; Array[a, 70] (* _Jean-François Alcover_, Dec 06 2015, adapted from PARI *)
%t Table[2*DivisorSigma[1,n]-DivisorSigma[0,n],{n,80}] (* _Harvey P. Dale_, Aug 07 2022 *)
%o (PARI) a(n)=sumdiv(2*n,d, if(d%2==0, d-1, 0 ) ); /* _Joerg Arndt_, Oct 07 2012 */
%o (PARI) a(n) = 2*sigma(n)-numdiv(n); \\ _Altug Alkan_, Mar 18 2018
%Y Cf. A129234, A129236, A129237.
%Y Cf. A000005, A000203.
%Y Cf. A130307.
%Y Cf. A051731, A143315, A143594.
%K nonn
%O 1,2
%A _Gary W. Adamson_, Apr 05 2007
%E Edited by _Emeric Deutsch_, Apr 17 2007
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