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A240991 Integers m such that A240923(m) = 1, where A240923(n) = numerator(sigma(n)/n) - sigma(denominator(sigma(n)/n)).

%I A240991 #63 Sep 01 2025 12:25:04
%S A240991 6,18,28,117,162,196,496,775,1458,8128,9604,13122,15376,19773,24025,
%T A240991 88723,118098,257049,470596,744775,796797,1032256,1062882,2896363,
%U A240991 6725201,9565938,12326221,14776336,23059204,25774633,27237961,33550336,43441281,63455131
%N A240991 Integers m such that A240923(m) = 1, where A240923(n) = numerator(sigma(n)/n) - sigma(denominator(sigma(n)/n)).
%C A240991 Perfect numbers (A000396) are a subsequence, since they satisfy sigma(m)/m = 2/1 = (sigma(1)+ 1)/1, that is of the form (sigma(d)+1)/d, with sigma being A000203.
%C A240991 Similarly, k-multiperfect numbers satisfy A240923(m) = k-1.
%C A240991 The analogous sequence of integers such that A240923(m) = 0 is A014567.
%C A240991 Holdener et al. say that these numbers have a quasi-friendly divisor and prove that such quasi-friendly divisors cannot have more than two distinct prime divisors. - _Michel Marcus_, Sep 08 2020, clarified by _Antti Karttunen_, Aug 31 2025
%C A240991 a(68) > 3.2*10^11. - _Giovanni Resta_, Aug 30 2025
%C A240991 Question: Might it possible to prove that for all n, A001221(a(n)) = 2, e.g., by showing that this is a subsequence of A387406? In any case, that holds for 67 initial terms. - _Antti Karttunen_, Aug 31 2025
%H A240991 Giovanni Resta, <a href="/A240991/b240991.txt">Table of n, a(n) for n = 1..67</a> (first 51 terms from Michel Marcus)
%H A240991 C. A. Holdener and J. A. Holdener, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Holdener/holdener4.html">Characterizing Quasi-Friendly Divisors</a>, Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.4.
%H A240991 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%H A240991 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%p A240991 filter:= proc(n) uses numtheory; local r; r:= sigma(n)/n; numer(r) - sigma(denom(r)) = 1 end proc:
%p A240991 select(filter, [$1..10^5]); # _Robert Israel_, Aug 07 2014
%t A240991 a240923[n_Integer] :=
%t A240991 Numerator[DivisorSigma[1, n]/n] -
%t A240991   DivisorSigma[1, Denominator[DivisorSigma[1, n]/n]];
%t A240991 a240991[n_Integer] := Flatten[Position[Thread[a240923[Range[n]]], 1]];
%t A240991 a240991[1000000] (* _Michael De Vlieger_, Aug 06 2014 *)
%Y A240991 Cf. A000203, A001221, A014567, A017665, A017666, A240923.
%Y A240991 Cf. also A387405 (the least k whose quasi-friendly divisor a(n) is, or -1 if no such k exists).
%Y A240991 Conjectured to be a subsequence of A387406.
%K A240991 nonn,changed
%O A240991 1,1
%A A240991 _Michel Marcus_, Aug 06 2014