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

%I A023897 #28 Dec 25 2024 04:24:49
%S A023897 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,
%T A023897 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,
%U A023897 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
%N A023897 a(n) = sigma_1(k) / phi(k) where k = A020492(n) is the n-th balanced number.
%C A023897 sigma_1(n) is the sum of the divisors of n [same as sigma(n)] (A000203).
%H A023897 Jud McCranie, <a href="/A023897/b023897.txt">Table of n, a(n) for n = 1..10000</a> (first 800 terms from Vincenzo Librandi)
%t A023897 Select[ Array[ DivisorSigma[ 1, # ]/EulerPhi[ # ]&, 20000 ], IntegerQ ]
%o A023897 (Magma) [ q: n in [1..20000] | r eq 0 where q, r is Quotrem(SumOfDivisors(n), EulerPhi(n)) ]; // _Klaus Brockhaus_, Nov 09 2008
%o A023897 (Python)
%o A023897 from math import prod
%o A023897 from itertools import count, islice
%o A023897 from sympy import factorint
%o A023897 def A023897_gen(startvalue=1): # generator of terms >= startvalue
%o A023897     for m in count(max(startvalue,1)):
%o A023897         f = factorint(m)
%o A023897         q, r = divmod(prod(p**(e+2)-p for p,e in f.items()),m*prod((p-1)**2 for p in f))
%o A023897         if not r:
%o A023897             yield q
%o A023897 A023897_list = list(islice(A023897_gen(),20)) # _Chai Wah Wu_, Aug 12 2024
%o A023897 (PARI) s(n) = {my(f = factor(n)); sigma(f)/eulerphi(f);}
%o A023897 list(lim) = select(x -> denominator(x) == 1, vector(lim, i, s(i))); \\ _Amiram Eldar_, Dec 25 2024
%Y A023897 Cf. A000010, A000203, A020492.
%K A023897 nonn
%O A023897 1,2
%A A023897 _Olivier Gérard_