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A180114
a(n) = sigma(A001694(n)), sum of divisors of the powerful number A001694(n).
8
1, 7, 15, 13, 31, 31, 40, 63, 91, 57, 127, 195, 121, 217, 280, 133, 156, 255, 403, 183, 399, 465, 600, 403, 364, 511, 819, 307, 847, 400, 381, 855, 961, 1240, 741, 931, 1092, 1023, 553, 1651, 781, 1815, 1240, 1281, 1093, 1767, 1953, 871, 2520, 2821, 993, 1995
OFFSET
1,2
LINKS
Rafael Jakimczuk and Matilde Lalín, 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. (7).
FORMULA
From Amiram Eldar, May 12 2023: (Start)
Sum_{A001694(k) < x} a(k) = c * x^(3/2) + O(x^(23/18 + eps)), where c = A362984 * A090699 / 3 = 1.5572721108... (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]
Sum_{k=1..n} a(k) ~ c * n^3, where c = A362984 / (3 * A090699^2) = 0.151716514097... . (End)
EXAMPLE
Sigma(2^2) = 7, sigma(2^3) = 15, sigma(3^2) = 13.
MAPLE
emin := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2], L); min(L) else 0 fi end: L:=[]: for w to 1 do for n from 1 to 144 do sn:=sigma(n); if emin(n)>1 then L:=[op(L), sn]; print(n, ifactor(n), sn, ifactor(sn)) fi; od; od;
MATHEMATICA
pwfQ[n_] := n == 1 || Min[Last /@ FactorInteger[n]] > 1; DivisorSigma[1, Select[ Range@ 1000, pwfQ]] (* Giovanni Resta, Feb 06 2018 *)
PROG
(PARI) lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 1, print1(sigma(k), ", "))); \\ Amiram Eldar, May 12 2023
(Python)
from itertools import count, islice
from math import prod
from sympy import factorint
def A180114_gen(): # generator of terms
for n in count(1):
f = factorint(n)
if all(e>1 for e in f.values()):
yield prod((p**(e+1)-1)//(p-1) for p, e in f.items())
A180114_list = list(islice(A180114_gen(), 20)) # Chai Wah Wu, May 21 2023
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot, divisor_sigma
def A180114(n):
def squarefreepi(n):
return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x):
c, l = n+x, 0
j = isqrt(x)
while j>1:
k2 = integer_nthroot(x//j**2, 3)[0]+1
w = squarefreepi(k2-1)
c -= j*(w-l)
l, j = w, isqrt(x//k2**3)
c -= squarefreepi(integer_nthroot(x, 3)[0])-l
return c
return divisor_sigma(bisection(f, n, n)) # Chai Wah Wu, Sep 10 2024
KEYWORD
easy,nonn
AUTHOR
Walter Kehowski, Aug 10 2010
EXTENSIONS
a(1)=1 prepended by and more terms from Giovanni Resta, Feb 06 2018
STATUS
approved