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A363194
Number of divisors of the n-th powerful number A001694(n).
7
1, 3, 4, 3, 5, 3, 4, 6, 9, 3, 7, 12, 5, 9, 12, 3, 4, 8, 15, 3, 9, 12, 16, 9, 6, 9, 18, 3, 15, 4, 3, 12, 15, 20, 9, 9, 12, 10, 3, 21, 5, 20, 12, 9, 7, 15, 18, 3, 24, 27, 3, 12, 18, 16, 11, 9, 12, 24, 9, 9, 25, 12, 4, 12, 3, 12, 9, 9, 18, 21, 3, 28, 27, 36, 3, 15
OFFSET
1,2
LINKS
Rafael Jakimczuk and Matilde LalĂ­n, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (8).
FORMULA
a(n) = A000005(A001694(n)).
Sum_{A001694(k) < x} a(k) = c_1 * sqrt(x) * log(x)^2 + c_2 * sqrt(x) * log(x) + c_3 * sqrt(x) + O(x^(5/12 + eps)), where c_1, c_2 and c_3 are constants. c_1 = Product_{p prime} (1 + 4/p^(3/2) - 1/p^2 - 6/p^(5/2) + 2/p^(7/2))/8 = 0.516273682988566836609... . [corrected Sep 21 2024]
a(n) = A343443(A306458(n)). - Amiram Eldar, Sep 01 2023
MATHEMATICA
DivisorSigma[0, Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]
PROG
(PARI) apply(numdiv, select(ispowerful, [1..3000]))
(Python)
from itertools import count, islice
from math import prod
from sympy import factorint
def A363194_gen(): # generator of terms
for n in count(1):
f = factorint(n).values()
if all(e>1 for e in f):
yield prod(e+1 for e in f)
A363194_list = list(islice(A363194_gen(), 20)) # Chai Wah Wu, May 21 2023
CROSSREFS
Similar sequences: A072048, A076400, A363195.
Sequence in context: A244055 A090739 A076400 * A121889 A332026 A205692
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, May 21 2023
STATUS
approved