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 A324543 Möbius transform of A323243, where A323243(n) = sigma(A156552(n)). 15
 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, 12, 27, 1023, 6, 2047, 8, 20, 48, 20, 20, 4095, 2, 78, 32, 8191, -6, 16383, 17, 9, 288, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The first three zeros after a(1) occur at n = 192, 288, 3645. There are 630 negative terms among the first 4473 terms. Applying this function to the divisors of the first four terms of A324201 reveals the following pattern: ---------------------------------------------------------------------------------- A324201(n) divisors                             a(n) applied to each:         Sum         9: [1, 3, 9]                         -> [0, 3, 9]                      12       125: [1, 5, 25, 125]                   -> [0, 7, 21, 28]                 56    161051: [1, 11, 121, 1331, 14641, 161051] -> [0, 31, 93, 124, 496, 248]    496 410338673: [1, 17, 289, 4913, 83521, 1419857, 24137569, 410338673]                              -> [0, 127, 381, 508, 2032, 1016, 9144, 3048]  16256. The second term (the first nonzero) of the latter list = A000668(n), and the sum is always twice the corresponding (even) 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. LINKS Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552) FORMULA a(n) = Sum_{d|n} A008683(n/d) * A323243(d). a(A000040(n)) = A000225(n). a(A001248(n)) = A173033(n) - A000225(n) = A068156(n) = 3*(2^n - 1). a(2*A000040(n)) = A324549(n). a(A002110(n)) = A324547(n). a(n) = 2*A297112(n) - A329644(n), and for n > 1, a(n) = 2^A297113(n) - A329644(n). - Antti Karttunen, Dec 08 2019 MATHEMATICA 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 *) 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)))); memoA323243 = Map(); A323243(n) = if(1==n, 0, my(v); if(mapisdefined(memoA323243, n, &v), v, v=sigma(A156552(n)); mapput(memoA323243, n, v); (v))); A324543(n) = sumdiv(n, d, moebius(n/d)*A323243(d)); CROSSREFS Cf. A000040, A000043, A000668, A000203, A000225, A000396, A008683, A068156, A156552, A173033, A297112, A297113, A323243, A323244, A324201, A324542, A324547, A324548, A324549, A324712, A329644. Sequence in context: A280753 A076217 A256784 * A333339 A089488 A076560 Adjacent sequences:  A324540 A324541 A324542 * A324544 A324545 A324546 KEYWORD sign AUTHOR Antti Karttunen, Mar 07 2019 STATUS approved

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Last modified January 15 16:11 EST 2021. Contains 340187 sequences. (Running on oeis4.)