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A324543 Möbius transform of A323243, where A323243(n) = sigma(A156552(n)). 12
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.

The function f(n) = 2*A297112(n) - a(n) is the Möbius transform of A323244.

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

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).

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, A323243, A323244, A324201, A324542, A324547, A324548, A324549, A324712.

Sequence in context: A280753 A076217 A256784 * A089488 A076560 A096915

Adjacent sequences:  A324540 A324541 A324542 * A324544 A324545 A324546

KEYWORD

sign

AUTHOR

Antti Karttunen, Mar 07 2019

STATUS

approved

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Last modified November 17 18:24 EST 2019. Contains 329241 sequences. (Running on oeis4.)