A297168 - OEIS

A297168

Difference between A156552 and its Moebius transform: a(n) = A156552(n) - A297112(n).

0, 0, 0, 1, 0, 3, 0, 3, 2, 5, 0, 7, 0, 9, 6, 7, 0, 9, 0, 11, 10, 17, 0, 15, 4, 33, 6, 19, 0, 17, 0, 15, 18, 65, 12, 19, 0, 129, 34, 23, 0, 29, 0, 35, 14, 257, 0, 31, 8, 17, 66, 67, 0, 21, 20, 39, 130, 513, 0, 35, 0, 1025, 22, 31, 36, 53, 0, 131, 258, 33, 0, 39, 0, 2049, 18, 259, 24, 101, 0, 47, 14, 4097, 0, 59, 68, 8193, 514, 71, 0, 37, 40

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OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8192

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = -Sum_{d|n, d<n} A008683(n/d)*A156552(d).

a(n) = Sum_{d|n, d<n} A297112(d).

For n > 1, a(n) = Sum_{d|n, 1<d<n} 2^A033265(A156552(d)).

a(n) = A156552(n) - A297112(n).

a(1) = 0, for n > 1, a(n) = A156552(n) - 2^A297167(n).

MATHEMATICA

With[{s = Array[Total@ MapIndexed[#1 2^(First@ #2 - 1) &, Flatten@ Map[ConstantArray[2^(PrimePi@ #1 - 1), #2] & @@ # &, FactorInteger@ #]] - Boole[# == 1]/2 &, 91]}, Table[-DivisorSum[n, MoebiusMu[n/#] s[[#]] &, # < n &], {n, Length@ s}]] (* Michael De Vlieger, Mar 13 2018 *)

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))));

A297112(n) = sumdiv(n, d, moebius(n/d)*A156552(d));

A297168(n) = (A156552(n)-A297112(n));

\\ Or alternatively as:

A297168(n) = -sumdiv(n, d, (d<n)*moebius(n/d)*A156552(d));

(PARI)

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));

A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));

A297112(n) = if(1==n, 0, 2^A297167(n));