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a(1) = 2; a(n) for n > 1 is the smallest k > a(n-1) such that the arithmetic mean of the divisors of k is one of the previous terms a(1), ..., a(n-1).
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%I #7 Sep 08 2022 08:45:47

%S 2,3,5,6,11,14,15,29,38,39,44,54,56,60,107,111,125,129,142,147,153,

%T 158,159,177,187,188,198,203,209,230,238,248,255,257,266,273,276,283,

%U 285,293,297,308,317,330,353,365,373,378,420,422,438,471,485,505,509,526

%N a(1) = 2; a(n) for n > 1 is the smallest k > a(n-1) such that the arithmetic mean of the divisors of k is one of the previous terms a(1), ..., a(n-1).

%C The arithmetic mean of the divisors of k is A000203(k)/A000005(k). a(n) for n > 1 is an arithmetic number, a term of A003601.

%C Similar sequences are obtained for other values of a(1). E.g. a(1) = 4 gives 4, 7, 13, 20, 45, 57, 89, 113, ... ; a(1) = 9 gives 9, 17, 22, 30, 43, 59, 87, 95, ... .

%e The smallest number with arithmetic mean of divisors = 2 is 3, hence a(2) = 3.

%e The next number with arithmetic mean of divisors in {2, 3} is 5, hence a(3) = 5.

%e The smallest number > a(5) = 11 with arithmetic mean of divisors in {a(1), ..., a(5)} = {2, 3, 5, 6, 11} is 14, hence a(6) = 14.

%o (Magma) S:={2}; for n in [1..600] do q, r:=Quotrem(&+Divisors(n), NumberOfDivisors(n)); if r eq 0 and q in S then Include(~S, n); end if; end for; S; // _Klaus Brockhaus_, Sep 05 2009

%Y Cf. A000005 (tau, sigma_0, number of divisors), A000203 (sigma, sum of divisors), A003601 (arithmetic mean of divisors is an integer, arithmetic numbers).

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Aug 27 2009

%E Edited and extended by _Klaus Brockhaus_, Sep 05 2009