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A297113 a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)). 12
0, 1, 2, 2, 3, 2, 4, 3, 3, 3, 5, 3, 6, 4, 3, 4, 7, 3, 8, 4, 4, 5, 9, 4, 4, 6, 4, 5, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 5, 13, 4, 14, 6, 4, 9, 15, 5, 5, 4, 7, 7, 16, 4, 5, 6, 8, 10, 17, 4, 18, 11, 5, 6, 6, 5, 19, 8, 9, 4, 20, 5, 21, 12, 4, 9, 5, 6, 22, 6, 5, 13, 23, 5, 7, 14, 10, 7, 24, 4, 6, 10, 11, 15, 8, 6, 25 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..12721

Index entries for sequences computed from indices in prime factorization

FORMULA

a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)) .

For n > 1, a(n) = A001511(A297112(n)), where A297112(n) = Sum_{d|n} moebius(n/d)*A156552(d).

a(n) = A252464(n) - A297155(n).

For n > 1, a(n) = 1+A033265(A156552(n)) = 1+A297167(n) = A046660(n) + A061395(n). - Last two sums added by Antti Karttunen, Sep 02 2018

Other identities. For all n >= 1:

a(A000040(n)) = n. [Each n occurs for the first time at the n-th prime.]

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)};

A297113(n) = if(n<=2, n-1, if(n%2, 1+A297113(A064989(n)), !(n%4)+A297113(n/2)));

\\ More complex way, after Moebius transform:

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));

A297113(n) = if(1==n, 0, 1+valuation(A297112(n), 2));

(Scheme, with memoization-macro definec)

(definec (A297113 n) (cond ((<= n 2) (- n 1)) ((= 2 (modulo n 4)) (A297113 (/ n 2))) (else (+ 1 (A297113 (A252463 n))))))

CROSSREFS

One more than A297167 (after the initial term).

Cf. A001511, A008683, A033265, A064989, A156552, A252463, A252464, A297112, A297155.

Cf. also A297157, A297161, A297162.

Sequence in context: A163870 A155043 A065770 * A086375 A107324 A023522

Adjacent sequences:  A297110 A297111 A297112 * A297114 A297115 A297116

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 26 2017

STATUS

approved

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Last modified February 17 17:27 EST 2019. Contains 320222 sequences. (Running on oeis4.)