OFFSET
1,2
COMMENTS
A number-theoretic sum involving the Euler phi function and gcd's.
Theorem 1.1 of Kiuchi and Tsuruta (2024) gives an estimate for a(n).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
I. Kiuchi and Y. Tsuruta, On sums involving the Euler totient function, Bull. Aust. Math. Soc. 109 (2024), 486-497.
EXAMPLE
For n = 3, the (a,b) pairs that appear in the sum are (1,1), (1,2), (1,3), (2,1), (3,1); the gcd of all is 1, and the sum of the phi-function at these 5 values of 1 is 5.
MAPLE
a:= proc(n) option remember; uses numtheory; `if`(n<1, 0,
a(n-1)+add(phi(igcd(d, n/d)), d=divisors(n)))
end:
seq(a(n), n=1..58); # Alois P. Heinz, Jul 04 2024
MATHEMATICA
a[n_] := a[n] = DivisorSum[n, EulerPhi[GCD[#, n/#]] &] + a[n - 1]; a[1] = 1; Array[a, 120] (* Michael De Vlieger, Jul 04 2024 *)
PROG
(Python)
from math import gcd
from functools import cache
from sympy import divisors, totient as phi
@cache
def a(n):
if n == 1: return 1
return a(n-1) + sum(phi(gcd(a, (b:=n//a))) for a in divisors(n))
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Jul 04 2024
(Python)
from math import prod
from itertools import count, islice
from sympy import factorint
def A372866_gen(): # generator of terms
c = 0
for n in count(1):
c += prod(p**(e>>1)+p**(e-1>>1) for p, e in factorint(n).items())
yield c
(PARI) a(n) = sum(k=1, n, sumdiv(k, d, eulerphi(gcd(d, k/d)))); \\ Michel Marcus, Jul 04 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, Jul 04 2024
STATUS
approved