A292385 -id:A292385 - cp's OEIS frontend

A292945 Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+5 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 5, 0, 10, 4, 0, 0, 21, 0, 42, 4, 4, 10, 85, 0, 0, 20, 0, 8, 171, 0, 342, 0, 8, 42, 1, 0, 684, 84, 20, 8, 1369, 8, 2738, 20, 0, 170, 5477, 0, 0, 0, 40, 40, 10955, 0, 8, 16, 84, 342, 21911, 0, 43822, 684, 8, 0, 17, 16, 87644, 84, 168, 2, 175289, 0, 350578, 1368, 0, 168, 3, 40, 701156, 16, 0, 2738, 1402313, 16, 40, 5476, 340, 40

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

The AND - XOR formulas are just a restatement of the fact that J(-3|n) = J(-1|n)*J(3|n), i.e., that Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292941, A292943, A292946.

Cf. also A292253, A292255.

(define (A292945 n) (if (<= n 1) 0 (+ (if (= 5 (modulo n 6)) 1 0) (* 2 (A292945 (A252463 n))))))

a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 5 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+5, and 0 otherwise.
Also, for n > 1, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(-3|n) = -1], where J is the Jacobi-symbol.
a(n) = A292263(n) AND (A292255(n) XOR A292383(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292263(n) AND (A292253(n) XOR A292385(n)). [See comments.]
For n >= 0, a(A163511(n)) = A292946(n).
For n >= 1, A292941(n) + A292943(n) + a(n) = A243071(n).

A292584 Compound filter: a(n) = P(A292583(n), A292585(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 2, 8, 5, 13, 2, 4, 8, 19, 5, 25, 13, 9, 2, 32, 4, 41, 8, 12, 19, 51, 5, 16, 25, 5, 13, 72, 9, 85, 2, 18, 32, 14, 4, 98, 41, 26, 8, 112, 12, 128, 19, 8, 51, 145, 5, 46, 16, 33, 25, 180, 5, 18, 13, 49, 72, 200, 9, 220, 85, 13, 2, 24, 18, 242, 32, 60, 14, 265, 4, 288, 98, 8, 41, 19, 26, 313, 8, 4, 112, 339, 12, 33, 128, 62, 19, 365, 8, 25, 51, 84, 145

Offset: 1

Antti Karttunen, Sep 20 2017

nonn

From Antti Karttunen, Sep 25 2017: (Start)
Some of the sequences this filter matches to:
For all i, j: a(i) = a(j) => A053866(i) = A053866(j).
For all i, j: a(i) = a(j) => A061395(i) = A061395(j). (also A006530, etc.)
For all i, j: a(i) = a(j) => A292378(i) = A292378(j).
The latter two implications follow simply because:
  A001222(A278222(A292383(n))) = A000120(A292383(n)) = A292377(n)
and, similarly, for n > 1,
  A001222(A278222(A292385(n))) = A000120(A292381(n)) = A292375(n),
and the sum of A292375(n) and A292377(n) is A061395(n) [index of the largest prime dividing n], while A292378 has been defined as 1 + their difference.
The case A053866 follows because of the component A292583, see comments under that entry. (End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A292383, A292385, A292583, A292585.

Cf. also A006530, A061395, A292378 (some of the matched sequences).

a(n) = (1/2)*(2 + ((A292583(n)+A292585(n))^2) - A292583(n) - 3*A292585(n)).

A292591 a(1) = 0, a(2) = 1; and for n > 2, a(n) = 2*a(A285712(n)) + [1 == (n mod 3)].

Original entry on oeis.org

0, 1, 2, 5, 2, 10, 21, 4, 42, 85, 10, 170, 5, 4, 340, 681, 20, 8, 1363, 42, 2726, 5453, 8, 10906, 11, 84, 21812, 21, 170, 43624, 87249, 20, 40, 174499, 340, 348998, 697997, 10, 16, 1395995, 8, 2791990, 85, 680, 5583980, 43, 1362, 168, 11167961, 40, 22335922, 44671845, 16, 89343690, 178687381, 2726, 357374762, 341, 84, 80, 23, 5452, 8, 714749525, 10906

Offset: 1

Antti Karttunen, Sep 20 2017

nonn

Binary expansion of a(n) encodes the positions of numbers of the form 3k+1 (with k >= 1) in the path taken from n to the root in the binary trees A245612 and A244154, except that the most significant 1-bit of a(n) always corresponds to 2 instead of 1 at the root of those trees.

Cf. A244153, A244154, A245611, A245612, A285712, A292590, A292593, A292595.

Cf. also A292245, A292385 (A292381).

f[n_] := f[n] = Which[n == 1, 0, Mod[n, 3] == 2, Ceiling[n/3], True, (Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1] + 1)/2]; a[n_] := a[n] = If[n <= 2, n - 1, 2 a[f@ n] + Boole[Mod[n, 3] == 1]]; Array[a, 65] (* Michael De Vlieger, Sep 22 2017 *)

(define (A292591 n) (if (<= n 2) (- n 1) (+ (if (= 1 (modulo n 3)) 1 0) (* 2 (A292591 (A285712 n))))))

a(n) + A292590(n) = A245611(n).
a(A245612(n)) = A292593(n).
A000120(a(n)) = A292595(n).

A332895 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].

Original entry on oeis.org

0, 1, 2, 2, 5, 4, 10, 4, 5, 10, 42, 8, 21, 20, 8, 8, 85, 10, 170, 20, 21, 84, 682, 16, 11, 42, 8, 40, 341, 16, 2730, 16, 85, 170, 16, 20, 1365, 340, 40, 40, 5461, 42, 10922, 168, 17, 1364, 43690, 32, 23, 22, 168, 84, 21845, 16, 80, 80, 341, 682, 174762, 32, 87381, 5460, 40, 32, 43, 170, 699050, 340, 1365, 32, 2796202

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

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

Cf. A000975, A108546, A332815, A332893, A332896, A332897.

Cf. also A292385.

A332895(n) = if(n<=2,n-1,2*A332895(A332893(n)) + (1==(n%4)));

a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].
For n > 1, a(2n) = 2*a(n).
For n >= 1, a(A108546(n)) = A000975(n); A000120(a(n)) = A332897(n).

cp's OEIS Frontend

A292945 Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+5 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

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A292584 Compound filter: a(n) = P(A292583(n), A292585(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A292591 a(1) = 0, a(2) = 1; and for n > 2, a(n) = 2*a(A285712(n)) + [1 == (n mod 3)].

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A332895 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].

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