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A300409
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Number of centered triangular numbers dividing n.
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6
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1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 2, 1, 2, 2, 1, 1, 1, 3
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OFFSET
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1,4
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LINKS
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FORMULA
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G.f.: Sum_{k>=0} x^(3*k*(k+1)/2+1)/(1 - x^(3*k*(k+1)/2+1)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A306324 = 1.5670651... . - Amiram Eldar, Jan 02 2024
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EXAMPLE
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a(20) = 3 because 20 has 6 divisors {1, 2, 4, 5, 10, 20} among which 3 divisors {1, 4, 10} are centered triangular numbers.
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MAPLE
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N:= 100: # for a(1)..a(N)
V:= Vector(N, 1):
for k from 1 do
m:= 3*k*(k+1)/2+1;
if m > N then break fi;
r:= [seq(i, i=m..N, m)];
V[r]:= map(t->t+1, V[r]);
od:
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MATHEMATICA
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nmax = 100; Rest[CoefficientList[Series[Sum[x^(3 k (k + 1)/2 + 1)/(1 - x^(3 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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