A064736 -id:A064736 - cp's OEIS frontend

A036552 List of pairs (m,2m) where m is the least unused positive number.

1, 2, 3, 6, 4, 8, 5, 10, 7, 14, 9, 18, 11, 22, 12, 24, 13, 26, 15, 30, 16, 32, 17, 34, 19, 38, 20, 40, 21, 42, 23, 46, 25, 50, 27, 54, 28, 56, 29, 58, 31, 62, 33, 66, 35, 70, 36, 72, 37, 74, 39, 78, 41, 82, 43, 86, 44, 88, 45, 90, 47, 94, 48, 96, 49, 98, 51, 102

Offset: 1

Tom Verhoeff

A permutation of the natural numbers. Inverse permutation is A065037.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. Erdős, R. Freud, and N. Hegyvári, Arithmetical properties of permutations of integers, Acta Mathematica Hungarica 41:1-2 (1983), pp. 169-176.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625[math.CO], 2020-2021.
Index entries for sequences that are permutations of the natural numbers

Alternating merge of A003159 and A036554. Cf. A064736, A065037.

import Data.List (delete)
a036552 n = a036552_list !! (n-1)
a036552_list = g [1..] where
   g (x:xs) = x : (2*x) : (g $ delete (2*x) xs)
-- Reinhard Zumkeller, Feb 07 2011

w = {}; Do[ w = If[ FreeQ[w, k], w = Join[w, {k, 2k}], w], {k, 100}]; w
(* Jean-François Alcover, Nov 04 2011, after Wouter Meeussen *)

apairs(N) = my(m=0, r=List(), i=0); while(#rRuud H.G. van Tol, May 09 2024

A064764 Largest integer m such that every permutation (p_1, ..., p_n) of (1, ..., n) satisfies lcm(p_i, p_{i+1}) >= m for some i, 1 <= i <= n-1.

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 12, 12, 12, 12, 18, 18, 24, 24, 24, 24, 35, 35, 44, 44, 44, 44, 55, 55, 55, 55, 55, 55, 68, 68, 85, 85, 85, 85, 85, 85, 102, 102, 102, 102, 119, 119, 145, 145, 145, 145, 174, 174, 174, 174, 174, 174, 203, 203, 203, 203, 203, 203, 232, 232, 261, 261, 261

Offset: 1

N. J. A. Sloane, Oct 21 2001

For n >= 4, a(n) >= A073818(pi(n)), with equality for 19 <= n <= 70. - David Wasserman, Aug 17 2002

			n=6: we must arrange the numbers 1..6 so that the max of the lcm of pairs of adjacent terms is minimized. The answer is 632415, with max lcm = 6, so a(6) = 6.

P. Erdős, R. Freud, and N. Hegyvári, Arithmetical properties of permutations of integers, Acta Mathematica Hungarica 41:1-2 (1983), pp 169-176.
D. Wasserman, Proof of terms 11-70

Cf. A035106, A064736, A064796, A064797, A000720, A073818.

a(n) = (1+o(1))n^2/(4 log n) as n -> infinity.

More terms from Vladeta Jovovic, Oct 21 2001

Further terms from David Wasserman, Aug 17 2002

A304085 Divisor-or-multiple permutation of natural numbers: a(n) = A052330(A304083(n)).

Original entry on oeis.org

1, 2, 6, 3, 24, 12, 4, 8, 120, 60, 20, 5, 40, 10, 30, 15, 840, 420, 140, 35, 7, 280, 70, 14, 210, 105, 21, 168, 84, 28, 56, 7560, 42, 1890, 945, 315, 63, 9, 3780, 1260, 252, 36, 2520, 630, 126, 18, 1512, 756, 189, 27, 378, 54, 1080, 540, 180, 45, 360, 90, 270, 135, 83160, 504, 72, 216, 108, 41580, 13860, 3465, 693, 99, 11, 27720, 6930, 1386, 198, 22, 20790

Offset: 0

Antti Karttunen, May 06 2018

nonn

Each a(n) is always either a divisor or a multiple of a(n+1).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A304086 (inverse).
Cf. A304083, A304087.
Cf. also A064736, A113552, A207901, A281978, A282291, A302350, A302781, A302783, A303751, A303771 for similar permutations.

up_to_e = 16; \\ Good for computing up to n = (2^16)-1
v050376 = vector(up_to_e);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A304085(n) = A052330(A304083(n)); \\ Needs also code from A304083

a(n) = A052330(A304083(n)).

A210770 a(1) = 1, a(2) = 2; for n > 1, a(2n+2) = smallest number not yet seen, a(2n+1) = a(2n) + a(2n+2).

Original entry on oeis.org

1, 2, 5, 3, 7, 4, 10, 6, 14, 8, 17, 9, 20, 11, 23, 12, 25, 13, 28, 15, 31, 16, 34, 18, 37, 19, 40, 21, 43, 22, 46, 24, 50, 26, 53, 27, 56, 29, 59, 30, 62, 32, 65, 33, 68, 35, 71, 36, 74, 38, 77, 39, 80, 41, 83, 42, 86, 44, 89, 45, 92, 47, 95, 48, 97, 49, 100

Offset: 1

Reinhard Zumkeller, Mar 25 2012

nonn

Permutation of natural numbers with inverse A210771.
From Jeffrey Shallit, Jun 18 2021: (Start)
This sequence is "2-sychronized"; there is a 23-state finite automaton that recognizes the base-2 representations of n and a(n), in parallel.
It obeys the identities
a(4n+3) = a(2n+1) - a(4n) + 2 a(4n+2)
a(8n) = 2a(4n)
a(8n+1) = a(2n+1) + 3a(4n)
a(8n+2) = a(2n+1) + 2 a(4n) - a(4n+1) + a(4n+2)
a(8n+4) = a(2n+1) + a(4n+2)
a(8n+5) = 3a(2n+1) - a(4n) +2a(4n+2)
a(8n+6) = 2a(2n+1) - a(4n) + a(4n+2). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A064736.

import Data.List (delete)
a210770 n = a210770_list !! (n-1)
a210770_list = 1 : 2 : f 1 2 [3..] where
   f u v (w:ws) = u' : w : f u' w (delete u' ws) where u' = v + w

def aupton(terms):
    alst, seen = [1, 2], {1, 2}
    for n in range(2, terms, 2):
        anp1 = alst[-1] + 1
        while anp1 in seen: anp1 += 1
        an = alst[n-1] + anp1
        alst, seen = alst + [an, anp1], seen | {an, anp1}
    return alst[:terms]
print(aupton(67)) # Michael S. Branicky, Jun 18 2021

a(2*n-1) = A022441(n-1); a(2*n) = A055562(n-1).

Definition corrected by Jeffrey Shallit, Jun 18 2021

A286293 A compressed version of A286290.

Original entry on oeis.org

1, 2, 4, 11, 17, 12, 14, 16, 29, 35, 24, 26, 28, 30, 32, 53, 59, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 98, 104, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 161, 167, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136

Offset: 1

N. J. A. Sloane, May 23 2017

nonn

			Start with A064736, bisect to get A286290, take second difference, and we get:
1, 2, 7, 6, -5, 2, 2, 13, 6, -11, 2, 2, 2, 2, 21, 6, -19, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 36, 6, -34, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 57, 6, -55, 2, 2, ...
which appears to consist of runs of 2's separated by a triple of numbers.
Look at the runs in that sequence:
1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 11, 1, 1, 1, 17, 1, 1, 1, 12, 1, 1, 1, 14, 1, 1, 1, 16, 1, 1, 1, 29, 1, 1, 1, 35, 1, 1, 1, 24, 1, 1, 1, 26, 1, 1, ...
and take the 4i+2 subsequence, which gives the present sequence.

Ray Chandler, Table of n, a(n) for n = 1..971

Cf. A064736, A286290, A286291, A286292.

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A036552 List of pairs (m,2m) where m is the least unused positive number.

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A064764 Largest integer m such that every permutation (p_1, ..., p_n) of (1, ..., n) satisfies lcm(p_i, p_{i+1}) >= m for some i, 1 <= i <= n-1.

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A304085 Divisor-or-multiple permutation of natural numbers: a(n) = A052330(A304083(n)).

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A210770 a(1) = 1, a(2) = 2; for n > 1, a(2*n+2) = smallest number not yet seen, a(2*n+1) = a(2*n) + a(2*n+2).

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A286293 A compressed version of A286290.

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A210770 a(1) = 1, a(2) = 2; for n > 1, a(2n+2) = smallest number not yet seen, a(2n+1) = a(2n) + a(2n+2).