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 A323244 a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)). 38
 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, 10, 1, 30, 14, 82, -2, 14, 1, 256, -12, 22, 1, 36, 1, 66, 8, 226, 1, 46, -12, 19, 8, 130, 1, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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. There are 2321 negative terms among the first 10000 terms. LINKS Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552) FORMULA a(n) = 2*A156552(n) - A323243(n). a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)). a(n) = A323248(n) + A001222(n) = (A323247(n) - A323243(n)) + A001222(n). From Antti Karttunen, Mar 12 2019 & Nov 23 2019: (Start) a(n) = Sum_{d|n} (2*A297112(d) - A324543(d)) = Sum_{d|n} A329644(d). A002487(a(n)) = A324115(n). a(n) = A329638(n) - A329639(n). a(n) = A329645(n) - A329646(n). (End) MATHEMATICA 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 *) PROG (PARI) 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)}; A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n)))); A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k)); CROSSREFS Cf. A000043, A000396, A033879, A064989, A156552, A297112, A323240, A323243, A323245, A323248, A324115, A324051, A324103, A324396, A324398, A324543, A324713. Cf. A324201 (positions of zeros, conjectured), A324551 (of negative terms), A324720 (of nonnegative terms), A324721 (of positive terms), A324731, A324732. Cf. A329644 (Möbius transform). Cf. A323174, A324055, A324185, A324546 for other permutations of deficiency, and also A324574, A324575, A324654. Sequence in context: A114326 A308175 A241423 * A329642 A214052 A276094 Adjacent sequences:  A323241 A323242 A323243 * A323245 A323246 A323247 KEYWORD sign AUTHOR Antti Karttunen, Jan 10 2019 STATUS approved

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Last modified June 14 00:02 EDT 2021. Contains 345016 sequences. (Running on oeis4.)