A292266 -id:A292266 - cp's OEIS frontend

A179680 The number of exponents >1 in a recursive reduction of 2n-1 until reaching an odd part equal to 1.

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

Offset: 1

Vladimir Shevelev, Jul 24 2010

nonn

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

			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.

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

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

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

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-François Alcover, Jul 30 2018, after R. J. Mathar *)

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

(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

A292265 A multiplicative encoding (compressed) for the exponents of 2 obtained when using Shevelev's algorithm for computing A002326.

Original entry on oeis.org

2, 3, 12, 6, 20, 180, 720, 5, 80, 25920, 20, 360, 43200, 25920, 6220800, 10, 240, 540, 671846400, 540, 57600, 2160, 540, 194400, 155520, 45, 5804752896000, 77760, 14400, 87071293440000, 348285173760000, 15, 960, 12538266255360000, 311040, 139968000, 120, 77760, 18662400, 1679616000, 23219011584000, 108330620446310400000, 60, 4665600, 360, 540, 180

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

a(n) = A019565(v(1)) * A019565(v(2)) * ... * A019565(v(k)), where v(1) .. v(k) are 2-adic valuations (not all necessarily distinct) of the iterated values obtained when running Shevelev's algorithm for computing A002623. (See A179680 and A292239.)

Antti Karttunen, Table of n, a(n) for n = 0..1023

Cf. A000265, A002326, A007814, A019565, A179680, A292239 (a variant), A292266 (rgs-version of this filter).

A000265(n) = (n >> valuation(n, 2));
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A292265(n) = { my(x = n+n+1, z = A019565(valuation(1+x,2)), m = A000265(1+x)); while(m!=1, z *= A019565(valuation(x+m,2)); m = A000265(x+m)); z; };

(define (A292265 n) (let ((x (+ n n 1))) (let loop ((z (A019565 (A007814 (+ 1 x)))) (k 1)) (let ((m (A000265 (+ x k)))) (if (= 1 m) z (loop (* z (A019565 (A007814 (+ x m)))) m))))))

For all n >= 0, A048675(a(n)) = A002326(n).

A293446 Restricted growth sequence transform of A293445, related to Shevelev's algorithm for computing A053446.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 3, 5, 6, 7, 3, 8, 9, 3, 10, 11, 12, 6, 13, 14, 15, 16, 17, 18, 19, 20, 21, 16, 22, 23, 24, 25, 26, 27, 13, 28, 27, 13, 29, 30, 23, 15, 17, 18, 31, 17, 32, 33, 34, 35, 20, 15, 36, 21, 16, 37, 17, 26, 23, 38, 13, 39, 40, 41, 42, 22, 27, 43, 44, 13, 28, 45, 46, 47, 18, 48, 49, 29, 50, 51, 52, 53, 54, 55, 56, 57, 34, 58, 35, 39, 59, 60, 61, 34

Offset: 1

Antti Karttunen, Oct 09 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..13122

Cf. A053446, A293445.

Cf. also A292266.

For all i, j: a(i) = a(j) => A053446(i) = A053446(j).

cp's OEIS Frontend

A179680 The number of exponents >1 in a recursive reduction of 2n-1 until reaching an odd part equal to 1.

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A292265 A multiplicative encoding (compressed) for the exponents of 2 obtained when using Shevelev's algorithm for computing A002326.

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A293446 Restricted growth sequence transform of A293445, related to Shevelev's algorithm for computing A053446.

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