A023897 a(n) = sigma_1(k) / phi(k) where k = A020492(n) is the n-th balanced number.

1, 3, 2, 6, 7, 4, 3, 9, 2, 8, 5, 6, 7, 4, 7, 10, 5, 12, 4, 9, 10, 3, 4, 14, 10, 8, 6, 13, 9, 8, 5, 15, 7, 2, 6, 8, 4, 5, 12, 6, 7, 10, 10, 11, 14, 12, 9, 4, 3, 4, 12, 9, 4, 4, 7, 5, 7, 10, 3, 5, 4, 13, 14, 12, 10, 9, 10, 8, 7, 4, 8, 6, 18, 9, 3, 8, 13, 8, 15, 15, 8, 3, 14, 9, 10, 8, 8, 10, 5, 7, 8, 11, 6, 11, 13, 6

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Comments

sigma_1(n) is the sum of the divisors of n [same as sigma(n)] (A000203).

Links

• Jud McCranie, Table of n, a(n) for n = 1..10000 (first 800 terms from Vincenzo Librandi)

Crossrefs

Cf. A000010, A000203, A020492.

Programs

Magma

[ q: n in [1..20000] | r eq 0 where q, r is Quotrem(SumOfDivisors(n), EulerPhi(n)) ]; // Klaus Brockhaus, Nov 09 2008

Mathematica

Select[ Array[ DivisorSigma[ 1, # ]/EulerPhi[ # ]&, 20000 ], IntegerQ ]

PARI

s(n) = {my(f = factor(n)); sigma(f)/eulerphi(f);}
list(lim) = select(x -> denominator(x) == 1, vector(lim, i, s(i))); \\ Amiram Eldar, Dec 25 2024

Python

from math import prod
from itertools import count, islice
from sympy import factorint
def A023897_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        f = factorint(m)
        q, r = divmod(prod(p**(e+2)-p for p,e in f.items()),m*prod((p-1)**2 for p in f))
        if not r:
            yield q
A023897_list = list(islice(A023897_gen(),20)) # Chai Wah Wu, Aug 12 2024