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A300409 Number of centered triangular numbers dividing n. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Eric Weisstein's World of Mathematics, Centered Triangular Number.
FORMULA
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
EXAMPLE
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.
MAPLE
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:
convert(V, list); # Robert Israel, Mar 05 2018
MATHEMATICA
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]]
CROSSREFS
Sequence in context: A105931 A349163 A279495 * A361631 A247462 A323172
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 05 2018
STATUS
approved

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Last modified March 29 09:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)