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Numbers n such that the average of the squares of the numbers less than n that do not divide n is an integer.
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%I #14 Aug 06 2024 21:58:29

%S 3,4,7,13,16,19,20,31,37,43,61,67,73,79,97,103,109,127,139,151,157,

%T 163,181,188,193,199,211,223,229,241,271,277,283,307,313,331,337,349,

%U 367,373,379,397,409,421,433,439,457,463,487,499,523,541,547,571,577,601,607,613,619,631,643,661,673,691,709,727,733,739,751,757,769,787

%N Numbers n such that the average of the squares of the numbers less than n that do not divide n is an integer.

%C Numbers n such that A049820(n) divides A276984(n).

%H Charles R Greathouse IV, <a href="/A279815/b279815.txt">Table of n, a(n) for n = 1..10000</a>

%e 7 is in the sequence because 7 has 2 divisors {1,7} therefore 5 non-divisors {2,3,4,5,6}, 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 90 and 5 divides 90.

%t Select[Range[800], Mod[#1 (#1 + 1) ((2 #1 + 1)/6) - DivisorSigma[2, #1], #1 - DivisorSigma[0, #1]] == 0 & ]

%o (PARI) is(n)=my(f=factor(n)); n>2 && ((2*n^3+3*n^2+n)/6-sigma(f,2))%(n-numdiv(f))==0 \\ _Charles R Greathouse IV_, Dec 19 2016

%Y Cf. A020486, A049820, A140826, A276984.

%Y Superset of A045375(m) (m > 1) ?

%K nonn,easy

%O 1,1

%A _Ilya Gutkovskiy_, Dec 19 2016