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A214547 Deficient numbers for which the (absolute value of) abundance is not a divisor.

%I A214547 #30 Apr 07 2024 09:09:23
%S A214547 3,5,7,9,11,13,14,15,17,19,21,22,23,25,26,27,29,31,33,34,35,37,38,39,
%T A214547 41,43,45,46,47,49,50,51,52,53,55,57,58,59,61,62,63,65,67,68,69,71,73,
%U A214547 74,75,76,77,79,81,82,83,85,86,87,89,91,92,93,94,95,97,98,99,101,103,105,106
%N A214547 Deficient numbers for which the (absolute value of) abundance is not a divisor.
%C A214547 This is to A214408 as deficient numbers are to abundant numbers.
%C A214547 Differs from A097218, which does not contain 105, for example.
%C A214547 The deficient numbers which are *not* in the sequence are 2, 4, 8, 10, 16, 32, 44, 64, 128, 136, 152, 184, 256, 512, 752, 884, 1024, 2048, 2144, 2272, 2528, 4096, 8192, 8384, 12224, 16384, 17176, 18632, 18904, 32768, 32896, 33664, ... the union of powers of 2 and the terms of A060326. - _M. F. Hasler_, Jul 21 2012
%H A214547 Robert Israel, <a href="/A214547/b214547.txt">Table of n, a(n) for n = 1..10000</a>
%F A214547 Terms A005100(n) such that |A033880(A005100(n))| does not divide A005100(n).
%e A214547 7 is in the sequence because 7 is deficient, and its abundance is -6, and |-6| = 6 does not divide 7.
%p A214547 filter:= proc(n) local t;
%p A214547 t:= 2*n-numtheory:-sigma(n);
%p A214547 t > 0 and n mod t <> 0
%p A214547 end proc:
%p A214547 select(filter, [$1..200]); # _Robert Israel_, Nov 13 2019
%t A214547 q[n_] := Module[{def = 2*n - DivisorSigma[1, n]}, def > 0 && !Divisible[n, def]]; Select[Range[120], q] (* _Amiram Eldar_, Apr 07 2024 *)
%o A214547 (PARI) is_A214547(n)={sigma(n)<2*n & n%(2*n-sigma(n))} \\ _M. F. Hasler_, Jul 21 2012
%Y A214547 Cf. A005100, A033880, A060326, A097218, A214408.
%K A214547 nonn,easy
%O A214547 1,1
%A A214547 _Jonathan Vos Post_, Jul 20 2012
%E A214547 Given terms double-checked with the PARI script by _M. F. Hasler_, Jul 21 2012