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A284288 Numbers n such that the average of the strong divisors of n is an integer.

%I A284288 #23 Apr 29 2021 12:33:03
%S A284288 2,3,4,5,7,9,11,13,17,19,23,25,27,28,29,31,37,41,43,47,49,53,54,56,59,
%T A284288 61,64,67,68,71,73,79,81,83,89,91,97,98,99,100,101,103,107,109,113,
%U A284288 121,127,131,133,137,138,139,148,149,151,154,157,163,165,167,169,173,179,181,188,191,192,193,197,199
%N A284288 Numbers n such that the average of the strong divisors of n is an integer.
%C A284288 We say d is a strong divisor of n iff d is a divisor of n and d > 1.
%C A284288 Numbers n such that A032741(n) divides A039653(n).
%C A284288 All primes and squares of primes are in this sequence.
%C A284288 Positions of ones in A296082 and A296084. - _Antti Karttunen_, Dec 05 2017
%H A284288 Robert Israel, <a href="/A284288/b284288.txt">Table of n, a(n) for n = 1..10000</a>
%H A284288 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%e A284288 28 is in the sequence because 28 has 6 divisors {1, 2, 4, 7, 14, 28} therefore 5 strong divisors {2, 4, 7, 14, 28}, 2 + 4 + 7 + 14 + 28 = 55 and 5 divides 55.
%p A284288 filter:= proc(n) local d,t;
%p A284288   d:= numtheory:-divisors(n) minus {1};
%p A284288   convert(d,`+`) mod nops(d) = 0
%p A284288 end proc:
%p A284288 select(filter, [$2..1000]); # _Robert Israel_, Mar 27 2017
%t A284288 Select[Range[2, 200], Mod[DivisorSigma[1, #1] - 1, DivisorSigma[0, #1] - 1] == 0 &]
%o A284288 (PARI) for(n=2, 200, if((sigma(n) - 1)%(numdiv(n) - 1)==0, print1(n,", "))) \\ _Indranil Ghosh_, Mar 24 2017
%o A284288 (Python)
%o A284288 from sympy.ntheory import divisor_sigma, divisor_count
%o A284288 print([n for n in range(2, 201) if (divisor_sigma(n) - 1)%(divisor_count(n) - 1) == 0]) # _Indranil Ghosh_, Mar 24 2017
%Y A284288 Cf. A000203, A000430, A003601, A023884, A023886, A032741, A039653, A296082, A296084 (characteristic function).
%K A284288 nonn
%O A284288 1,1
%A A284288 _Ilya Gutkovskiy_, Mar 24 2017