A353551 a(n) = Sum_{k=1..n} tau(k^3), where tau is the number of divisors function A000005.
0, 1, 5, 9, 16, 20, 36, 40, 50, 57, 73, 77, 105, 109, 125, 141, 154, 158, 186, 190, 218, 234, 250, 254, 294, 301, 317, 327, 355, 359, 423, 427, 443, 459, 475, 491, 540, 544, 560, 576, 616, 620, 684, 688, 716, 744, 760, 764, 816, 823, 851, 867, 895, 899, 939, 955, 995
Offset: 0
Keywords
Examples
A048785(0) = 0 + A048785(1) = 1 + A048785(2) = 4 + A048785(3) = 4 ------------------ = A353551(3) = 9
Links
- Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000
Programs
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Maple
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+numtheory[tau](n^3)) end: seq(a(n), n=0..100); # Alois P. Heinz, May 08 2022
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Mathematica
Accumulate[Join[{0}, Table[DivisorSigma[0, k^3], {k, 1, 50}]]] (* Amiram Eldar, May 08 2022 *) -
PARI
a(n) = sum(k=1, n, numdiv(k^3)); \\ Michel Marcus, May 08 2022
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Python
from sympy import divisor_count def A048785(n): return divisor_count(n**3) def A353551(n): return sum(A048785(n) for n in range(1, n)) print([A353551(n) for n in range(1, 58)])
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Python
from math import prod from sympy import factorint def A353551(n): return sum(prod(3*e+1 for e in factorint(k).values()) for k in range(1,n+1)) # Chai Wah Wu, May 10 2022
Formula
a(n) = Sum_{k=1..n} tau(k^3).
a(n) = a(n-1) + A048785(n) for n >= 1, a(0) = 0.