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A297111 Möbius transform of A005187, where A005187(n) = 2n - (number of 1's in binary representation of n). 13
1, 2, 3, 4, 7, 4, 10, 8, 12, 8, 18, 8, 22, 12, 15, 16, 31, 12, 34, 16, 25, 20, 41, 16, 39, 24, 34, 24, 53, 16, 56, 32, 42, 32, 49, 24, 70, 36, 48, 32, 78, 24, 81, 40, 48, 44, 88, 32, 84, 40, 63, 48, 101, 36, 79, 48, 72, 56, 112, 32, 116, 60, 69, 64, 98, 40, 130, 64, 90, 48, 137, 48, 142, 72, 81, 72, 121, 48, 152, 64 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sequence differs from A035532 for the first time at n = 15, 21, 25, 27, 33, 35, 51, etc., i.e., at those composite n where A297115 has a nonzero value. - Antti Karttunen & M. F. Hasler, Mar 10 2018

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

FORMULA

a(n) = Sum_{d|n} A005187(d)*A008683(n/d).

a(n) = n + A297114(n).

From Antti Karttunen, Mar 11 2018: (Start)

Sum A005187(n) x^n = Sum a(n)*x^n/(1-x^n). [Another way of saying that this is the Möbius transform of A005187. This was originally included in A035532 by mistake.]

a(n) = 2*phi(n) - A297115(n) = phi(n) + A297117(n).

a(n) = A005187(n) - A300244(n).

a(1) = 1; for n > 1, a(n) = A300723(n) + 2*A300724(n).

(End)

MATHEMATICA

Table[DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 80}] (* Michael De Vlieger, Mar 10 2018 *)

PROG

(PARI)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };

A297111(n) = sumdiv(n, d, moebius(n/d)*A005187(d));

CROSSREFS

Cf. A000010, A005187, A008683, A297108, A297110, A297114, A297115, A297117, A300244, A300723, A300724, A300725.

Sequence in context: A297110 A267695 A138676 * A035532 A176535 A251716

Adjacent sequences:  A297108 A297109 A297110 * A297112 A297113 A297114

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 25 2017

STATUS

approved

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Last modified October 19 16:17 EDT 2019. Contains 328223 sequences. (Running on oeis4.)