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%I #47 Mar 15 2019 12:34:27
%S 1,1,2,1,4,0,5,1,6,2,6,-4,19,-3,14,1,10,4,10,-2,22,-12,12,-12,41,7,26,
%T -19,94,-12,41,1,12,8,18,0,38,-12,22,-10,58,-4,18,-48,102,-54,30,-28,
%U 109,25,66,-17,148,-72,47,-51,286,32,126,-64,469,-39,122,1,16,10,22,4,46,-12,42,-8,70,4,42,-56,178,-60,58,-26,118,20
%N Deficiency of Doudna-sequence: a(n) = A033879(A005940(1+n)).
%C Both here and in the mirror image sequence A324185, the lowermost (asinh) scatter plot shows on the y = 0 line the numbers that correspond to the perfect numbers. Compare also to the scatter plot of A243492.
%H Antti Karttunen, <a href="/A324055/b324055.txt">Table of n, a(n) for n = 0..8192</a>
%H Antti Karttunen, <a href="/A324055/a324055.txt">Data supplement: n, a(n) computed for n = 0..70327</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(n) = A033879(A005940(1+n)).
%F a(n) = 2*A005940(1+n) - A324054(n).
%F For n > 0, a(n) = A324185(A054429(n)).
%F a(n) = A324348(n) + A000120(A005940(1+n)).
%t Array[Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2]}, 2 # - DivisorSigma[1, #] &[Times @@ Flatten@ Table[Prime[Count[Flatten@ #, 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]] &, 82, 0] (* _Michael De Vlieger_, Mar 11 2019, after _Robert G. Wilson v_ at A005940 *)
%o (PARI)
%o A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
%o A033879(n) = (2*n-sigma(n));
%o A324055(n) = A033879(A005940(1+n));
%o (PARI) A324055(n) = { my(m1=2,m2=1,p=2,mp=p*p); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, m1 *= p; if(3==(n%4),mp *= p,m2 *= (mp-1)/(p-1))); n>>=1); (m1-m2); };
%Y Cf. A000120, A033879, A054429.
%Y See A106737, A290077, A323915, A324052, A324054, A324056, A324057, A324058, A324114, A324335, A324340, A324348, A324349, A324394, A324395 for other sequences as permuted by A005940, and compare their scatter plots.
%Y Cf. also A243492, A323174, A323244, A324185, A324346, A324347, A324654.
%K sign,look
%O 0,3
%A _Antti Karttunen_, Feb 14 2019
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