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Integers m such that A240923(m) = 1.
1

%I #37 Sep 14 2020 17:29:39

%S 6,18,28,117,162,196,496,775,1458,8128,9604,13122,15376,19773,24025,

%T 88723,118098,257049,470596,744775,796797,1032256,1062882,2896363,

%U 6725201,9565938,12326221,14776336,23059204,25774633,27237961,33550336,43441281,63455131

%N Integers m such that A240923(m) = 1.

%C 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 Similarly, k-multiperfect numbers satisfy A240923(m) = k-1.

%C The analogous sequence of integers such that A240923(m) = 0 is A014567.

%C Holdener et al. say that these numbers have a quasi-friendly divisor and prove that they cannot have more than two distinct prime divisors. - _Michel Marcus_, Sep 08 2020

%H Michel Marcus, <a href="/A240991/b240991.txt">Table of n, a(n) for n = 1..51</a>

%H 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.

%p filter:= proc(n) uses numtheory; local r; r:= sigma(n)/n; numer(r) - sigma(denom(r)) = 1 end proc:

%p select(filter, [$1..10^5]); # _Robert Israel_, Aug 07 2014

%t a240923[n_Integer] :=

%t Numerator[DivisorSigma[1, n]/n] -

%t DivisorSigma[1, Denominator[DivisorSigma[1, n]/n]];

%t a240991[n_Integer] := Flatten[Position[Thread[a240923[Range[n]]], 1]];

%t a240991[1000000] (* _Michael De Vlieger_, Aug 06 2014 *)

%Y Cf. A000203, A014567, A017665, A017666.

%K nonn

%O 1,1

%A _Michel Marcus_, Aug 06 2014