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A324543 Möbius transform of A323243, where A323243(n) = sigma(A156552(n)).

%I A324543 #59 Nov 07 2024 15:41:30
%S A324543 0,1,3,3,7,2,15,4,9,5,31,3,63,2,8,16,127,-1,255,4,21,16,511,8,21,20,
%T A324543 12,27,1023,6,2047,8,20,48,20,20,4095,2,78,32,8191,-6,16383,17,9,288,
%U A324543 32767,8,45,-3,122,45,65535,4,53,20,270,278,131071,2,262143,688,12,72,56,23,524287,125,260,-8,1048575,20,2097151,260,3,363,44,-7,4194303
%N A324543 Möbius transform of A323243, where A323243(n) = sigma(A156552(n)).
%C A324543 The first four zeros after a(1) occur at n = 192, 288, 3645, 6075.
%C A324543 There are 1562 negative terms among the first 10000 terms.
%C A324543 Applying this function to the divisors of the first four terms of A324201 reveals the following pattern:
%C A324543 ----------------------------------------------------------------------------------
%C A324543   A324201  divisors                             a(n) applied to each:         Sum
%C A324543         9: [1, 3, 9]                         -> [0, 3, 9]                      12 = 2*6
%C A324543       125: [1, 5, 25, 125]                   -> [0, 7, 21, 28]                 56 = 2*28
%C A324543    161051: [1, 11, 121, 1331, 14641, 161051] -> [0, 31, 93, 124, 496, 248]    992 = 2*496
%C A324543 410338673: [1, 17, 289, 4913, 83521, 1419857, 24137569, 410338673]
%C A324543                              -> [0, 127, 381, 508, 2032, 1016, 9144, 3048]  16256 = 2*8128
%C A324543 The second term (the first nonzero) of the latter list = A000668(n), and the sum is always twice the corresponding perfect number, which forces either it or at least many of its divisors to be present. For example, in the fourth case, although 8128 = A000396(4) itself is not present, we still have 127, 508, 1016 and 2032 in the list. See also A329644.
%H A324543 Antti Karttunen, <a href="/A324543/b324543.txt">Table of n, a(n) for n = 1..10000</a> (based on Hans Havermann's factorization of A156552)
%H A324543 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A324543 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A324543 a(n) = Sum_{d|n} A008683(n/d) * A323243(d).
%F A324543 a(A000040(n)) = A000225(n).
%F A324543 a(A001248(n)) = A173033(n) - A000225(n) = A068156(n) = 3*(2^n - 1).
%F A324543 a(2*A000040(n)) = A324549(n).
%F A324543 a(A002110(n)) = A324547(n).
%F A324543 a(n) = 2*A297112(n) - A329644(n), and for n > 1, a(n) = 2^A297113(n) - A329644(n). - _Antti Karttunen_, Dec 08 2019
%t A324543 Table[DivisorSum[n, MoebiusMu[n/#] If[# == 1, 0, DivisorSigma[1, Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]]]] &], {n, 79}] (* _Michael De Vlieger_, Mar 11 2019 *)
%o A324543 (PARI)
%o A324543 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A324543 A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
%o A324543 memoA323243 = Map();
%o A324543 A323243(n) = if(1==n, 0, my(v); if(mapisdefined(memoA323243,n,&v),v, v=sigma(A156552(n)); mapput(memoA323243,n,v); (v)));
%o A324543 A324543(n) = sumdiv(n,d,moebius(n/d)*A323243(d));
%Y A324543 Cf. A000040, A000043, A000668, A000203, A000225, A000396, A008683, A068156, A156552, A173033, A297112, A297113, A323243, A323244, A324201, A324542, A324547, A324548, A324549, A324712, A329644.
%K A324543 sign
%O A324543 1,3
%A A324543 _Antti Karttunen_, Mar 07 2019