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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
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OFFSET
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1,3
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COMMENTS
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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:
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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.
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LINKS
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FORMULA
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MATHEMATICA
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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 *)
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PROG
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(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)};
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)));
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CROSSREFS
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Cf. A000040, A000043, A000668, A000203, A000225, A000396, A008683, A068156, A156552, A173033, A297112, A297113, A323243, A323244, A324201, A324542, A324547, A324548, A324549, A324712, A329644.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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