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Let s(m) = Sum_{j=1..m} sigma(j)/j where sigma(j) is the sum of divisors of j. Sequence consists of numbers k such that denominator(s(k)) < denominator(s(k-1)).
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%I #24 Feb 14 2021 13:01:15

%S 10,50,55,69,77,102,184,185,186,238,250,261,268,322,483,522,605,620,

%T 671,693,774,847,1159,1204,1222,1250,1316,1507,1548,1587,1734,1870,

%U 1880,1938,2032,2055,2212,2409,2440,2575,2653,3082,3399,3416,3448,3685,4033,4046

%N Let s(m) = Sum_{j=1..m} sigma(j)/j where sigma(j) is the sum of divisors of j. Sequence consists of numbers k such that denominator(s(k)) < denominator(s(k-1)).

%C Overall the sequence defined by s(m) = Sum_{j=1..m} sigma(j)/j has increasing denominators, but there are some values of m where the denominators locally decrease.

%H Jens Kruse Andersen, <a href="/A243226/b243226.txt">Table of n, a(n) for n = 1..1000</a>

%e The first 10 terms of {s(n)} are 1, 5/2, 23/6, 67/12, 407/60, 527/60, 4169/420, 9913/840, 33379/2520, 7583/504.

%e Since denominator(s(10)) < denominator(s(9)), 10 is in the sequence.

%p N:= 10^4: # to get all terms <= N

%p tot:= 0:

%p for n from 1 to N+1 do tot:= tot+numtheory:-sigma(n)/n; sd[n]:= denom(tot) od:

%p select(t -> sd[t] < sd[t-1],[$2..N]); # _Robert Israel_, Aug 22 2014

%o (PARI) lista(nn) = {s = 1; lastd = 1; for (n=2, nn, s += sigma(n)/n; newd = denominator(s); if (newd < lastd, print1(n, ", ");); lastd = newd;);}

%Y Cf. A000203, A017665, A017666.

%K nonn

%O 1,1

%A _Michel Marcus_, Aug 21 2014