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Sum of centered triangular numbers dividing n.
1

%I #11 May 19 2020 18:25:17

%S 1,1,1,5,1,1,1,5,1,11,1,5,1,1,1,5,1,1,20,15,1,1,1,5,1,1,1,5,1,11,32,5,

%T 1,1,1,5,1,20,1,15,1,1,1,5,1,47,1,5,1,11,1,5,1,1,1,5,20,1,1,15,1,32,1,

%U 69,1,1,1,5,1,11,1,5,1,1,1,24,1,1,1,15,1,1,1,5,86,1,1,5,1,11

%N Sum of centered triangular numbers dividing n.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredTriangularNumber.html">Centered Triangular Number</a>

%F G.f.: Sum_{k>=1} (3*k*(k - 1)/2 + 1) * x^(3*k*(k - 1)/2 + 1) / (1 - x^(3*k*(k - 1)/2 + 1)).

%F L.g.f.: log(G(x)), where G(x) is the g.f. for A280950.

%t nmax = 90; CoefficientList[Series[Sum[(3 k (k - 1)/2 + 1) x^(3 k (k - 1)/2 + 1)/(1 - x^(3 k (k - 1)/2 + 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t nmax = 90; CoefficientList[Series[Log[Product[1/(1 - x^(3 k (k - 1)/2 + 1)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

%o (PARI) isc(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k));

%o a(n) = sumdiv(n, d, if (isc(d), d)); \\ _Michel Marcus_, May 19 2020

%Y Cf. A005448, A185027, A280950, A300409, A334988.

%K nonn

%O 1,4

%A _Ilya Gutkovskiy_, May 18 2020