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A179680 The number of exponents >1 in a recursive reduction of 2n-1 until reaching an odd part equal to 1. 7
0, 1, 1, 1, 1, 3, 3, 1, 1, 5, 1, 3, 5, 5, 7, 1, 1, 3, 9, 3, 3, 3, 3, 6, 5, 2, 13, 5, 3, 15, 15, 1, 1, 17, 5, 9, 1, 5, 7, 10, 13, 21, 1, 7, 2, 3, 2, 9, 11, 9, 25, 13, 2, 27, 9, 9, 5, 11, 2, 6, 27, 5, 25, 1, 1, 33, 3, 9, 15, 35, 11, 15, 3, 11, 37, 3, 6, 5, 13, 13 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Let N = 2n-1. Then consider the following algorithm of updating pairs (v,m) indicating highest exponent of 2 (2-adic valuation) and odd part: Initialize at step 1 by v(1) = A007814(N+1) and m(1) = A000265(N+1). Iterate over steps i>=2: v(i) = A007814(N+m(i-1)), m(i) = A000265(N+m(i-1)) using the previous odd part m(i-1) until some m(k) = 1. a(n) is defined as the count of the v(i) which are larger than 1.

This is an algorithm to compute A002326 because the sum v(1)+v(2)+ ... +v(k) of the exponents is A002326(n-1).

A179382(n) = 1 + the number of iterations taken by the algorithm when starting from N = 2n-1. - Antti Karttunen, Oct 02 2017

LINKS

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

EXAMPLE

For n = 9, 2*n-1 = 17, we have v_1 = v_2 = v_3 = 1, v_4 = 5. Thus a(9) = 1.

For n = 10, 2*n-1 = 19, we have v_1 = 2, v_2 = 3, v_3 = v_4 = v_5 = 1, v_6 = v_7 = 2, v_8 = 1, v_9 = 5. Thus a(10) = 5.

MAPLE

A179680 := proc(n) local l, m, a , N ; N := 2*n-1 ; a := 0 ; l := A007814(N+1) ; m := A000265(N+1) ; if l > 1 then a := a+1 ; end if; while m <> 1 do l := A007814(N+m) ; if l > 1 then a := a+1 ; end if; m := A000265(N+m) ; end do: a ; end proc:

seq(A179680(n), n=1..80) ; # R. J. Mathar, Apr 05 2011

MATHEMATICA

a7814[n_] := IntegerExponent[n, 2];

a265[n_] := n/2^IntegerExponent[n, 2];

a[n_] := Module[{l, m, k, nn}, nn = 2n-1; k = 0; l = a7814[nn+1]; m = a265[nn+1]; If[l>1, k++]; While[m != 1, l = a7814[nn+m]; If[l>1, k++]; m = a265[nn+m]]; k];

Array[a, 80] (* Jean-Fran├žois Alcover, Jul 30 2018, after R. J. Mathar *)

PROG

(Scheme) (define (A179680 n) (let ((x (+ n n -1))) (let loop ((s (- 1 (A000035 n))) (k 1)) (let ((m (A000265 (+ x k)))) (if (= 1 m) s (loop (+ s (if (> (A007814 (+ x m)) 1) 1 0)) m)))))) ;; Antti Karttunen, Oct 02 2017

(Sage)

def A179680(n):

    s, m, N = 0, 1, 2*n - 1

    while True:

        k = N + m

        v = valuation(k, 2)

        if v > 1: s += 1

        m = k >> v

        if m == 1: break

    return s

print([A179680(n) for n in (1..80)]) # Peter Luschny, Oct 07 2017

CROSSREFS

Cf. A179676, A179460, A007814, A002326, A179382, A292239, A292265, A292266.

Sequence in context: A296523 A171876 A133332 * A123562 A046218 A046221

Adjacent sequences:  A179677 A179678 A179679 * A179681 A179682 A179683

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Jul 24 2010

STATUS

approved

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Last modified October 15 05:31 EDT 2018. Contains 316202 sequences. (Running on oeis4.)