|
|
A372866
|
|
a(n) is the sum, over all positive integers x, y such that x*y <= n, of phi(gcd(x,y)).
|
|
2
|
|
|
1, 3, 5, 8, 10, 14, 16, 20, 24, 28, 30, 36, 38, 42, 46, 52, 54, 62, 64, 70, 74, 78, 80, 88, 94, 98, 104, 110, 112, 120, 122, 130, 134, 138, 142, 154, 156, 160, 164, 172, 174, 182, 184, 190, 198, 202, 204, 216, 224, 236, 240, 246, 248, 260, 264, 272, 276, 280
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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:
|
|
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))
(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
|
|
|
STATUS
|
approved
|
|
|
|