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sigma_1(n) / phi(n) for balanced numbers.
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%I #24 Aug 12 2024 12:02:25

%S 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 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 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 sigma_1(n) / phi(n) for balanced numbers.

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

%H Jud McCranie, <a href="/A023897/b023897.txt">Table of n, a(n) for n = 1..10000</a> (first 800 terms from Vincenzo Librandi)

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

%o (Magma) [ q: n in [1..20000] | r eq 0 where q, r is Quotrem(SumOfDivisors(n), EulerPhi(n)) ]; // _Klaus Brockhaus_, Nov 09 2008

%o (Python)

%o from math import prod

%o from itertools import count, islice

%o from sympy import factorint

%o def A023897_gen(startvalue=1): # generator of terms >= startvalue

%o for m in count(max(startvalue,1)):

%o f = factorint(m)

%o q, r = divmod(prod(p**(e+2)-p for p,e in f.items()),m*prod((p-1)**2 for p in f))

%o if not r:

%o yield q

%o A023897_list = list(islice(A023897_gen(),20)) # _Chai Wah Wu_, Aug 12 2024

%Y Cf. A000010, A000203, A020492.

%K nonn

%O 1,2

%A _Olivier GĂ©rard_