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

Offset: 1

Author

Antti Karttunen, Feb 27 2018

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A008683, A033265, A156552, A297112, A297112, A297113, A297167, A297169, A300827.

Programs

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 *)

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,(dA156552(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));
A297168(n) = sumdiv(n,d,(dA297112(d)); \\ Antti Karttunen, Mar 13 2018

Scheme

(define (A297168 n) (- (A156552 n) (A297112 n)))
(define (A297168 n) (if (= 1 n) 0 (- (A156552 n) (A000079 (A297167 n)))))

Formula

a(n) = -Sum_{d|n, dA008683(n/d)*A156552(d).

a(n) = Sum_{d|n, dA297112(d).

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

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

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