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%I #43 Mar 11 2023 08:08:26
%S 0,1,1,2,1,4,1,6,0,5,1,10,1,16,2,6,1,12,1,18,-3,18,1,22,-4,46,4,22,1,
%T 10,1,30,14,82,-2,14,1,256,-12,22,1,36,1,66,8,226,1,46,-12,19,8,130,1,
%U 28,-19,70,-12,748,1,42,1,1362,16,22,10,42,1,214,254,40,1,38,1,3838,10,406,-10,106,1,78,-12,5458,1,26,-72,12250,-348,30,1,12
%N a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).
%C After a(1) = 0, the other zeros occur for k >= 1, at A005940(1+A000396(k)), which, provided no odd perfect numbers exist, is equal to A324201(k) = A062457(A000043(k)): 9, 125, 161051, 410338673, ..., etc.
%C There are 2321 negative terms among the first 10000 terms.
%H Antti Karttunen, <a href="/A323244/b323244.txt">Table of n, a(n) for n = 1..10000</a> (based on Hans Havermann's factorization of A156552)
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(n) = 2*A156552(n) - A323243(n).
%F a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).
%F a(n) = A323248(n) + A001222(n) = (A323247(n) - A323243(n)) + A001222(n).
%F From _Antti Karttunen_, Mar 12 2019 & Nov 23 2019: (Start)
%F a(n) = Sum_{d|n} (2*A297112(d) - A324543(d)) = Sum_{d|n} A329644(d).
%F A002487(a(n)) = A324115(n).
%F a(n) = A329638(n) - A329639(n).
%F a(n) = A329645(n) - A329646(n).
%F (End)
%t Array[2 # - If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 90] (* _Michael De Vlieger_, Apr 21 2019 *)
%o (PARI)
%o 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 A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
%o A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));
%o (Python)
%o from sympy import divisor_sigma, primepi, factorint
%o def A323244(n): return (lambda n: (n<<1)-divisor_sigma(n))(sum((1<<primepi(p)-1)<<i for i, p in enumerate(factorint(n,multiple=True)))) if n > 1 else 0 # _Chai Wah Wu_, Mar 10 2023
%Y Cf. A000043, A000396, A033879, A064989, A156552, A297112, A323240, A323243, A323245, A323248, A324115, A324051, A324103, A324396, A324398, A324543, A324713.
%Y Cf. A324201 (positions of zeros, conjectured), A324551 (of negative terms), A324720 (of nonnegative terms), A324721 (of positive terms), A324731, A324732.
%Y Cf. A329644 (Möbius transform).
%Y Cf. A323174, A324055, A324185, A324546 for other permutations of deficiency, and also A324574, A324575, A324654.
%K sign
%O 1,4
%A _Antti Karttunen_, Jan 10 2019
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