A340510 A permutation of the positive integers with a divisibility property (see Comments for precise definition).

1, 3, 5, 2, 8, 10, 4, 13, 15, 6, 18, 7, 21, 23, 9, 26, 28, 11, 31, 12, 34, 36, 14, 39, 41, 16, 44, 17, 47, 49, 19, 52, 20, 55, 57, 22, 60, 62, 24, 65, 25, 68, 70, 27, 73, 75, 29, 78, 30, 81, 83, 32, 86, 33, 89, 91, 35, 94, 96, 37, 99, 38, 102, 104, 40, 107, 109

Offset

1,2

Comments

a(1)=1; thereafter a(n) is the least positive number not yet in the sequence such that Sum_{i=1..n} a(i) == 1 mod n+1.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.

J. Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], August 12 2023.

Formula

Theorem 1 of Avdispahić and Zejnulahi gives an explicit formula involving Fibonacci numbers.

Maple

A000045_inv := proc(n)
    local k ;
    for k from 0 do
        if combinat[fibonacci](k) = n then
            return k;
        elif combinat[fibonacci](k) > n then
            return -1;
        end if;
    end do:
end proc:
A000071_inv := proc(n)
    local k ;
    for k from 0 do
        if combinat[fibonacci](k)-1 = n then
            return k;
        elif combinat[fibonacci](k)-1 > n then
            return -1;
        end if;
    end do:
end proc:
A000201_inv := proc(n)
    local k, w ;
    for k from 1 do
        w := floor(k*(1+sqrt(5))/2) ;
        if w = n then
            return k;
        elif w > n then
            return -1;
        end if;
    end do:
end proc:
A001950_inv := proc(n)
    local k, w ;
    for k from 1 do
        w := floor(k*(3+sqrt(5))/2) ;
        if w = n then
            return k;
        elif w > n then
            return -1;
        end if;
    end do:
end proc:
A340510 := proc(n)
    local k ;
    if n = 1 then
        1;
    else
        k := A000045_inv(n) ;
        if k > 2 then
            return combinat[fibonacci](k+1) ;
        end if;
        k := A000071_inv(n) ;
        if k > 4 then
            return combinat[fibonacci](k-1)-1 ;
        end if;
        k := A000201_inv(n) ;
        if k > 0 then
            return floor(k*(3+sqrt(5))/2) ;
        end if;
        k := A001950_inv(n) ;
        return floor(k*(1+sqrt(5))/2) ;
    end if;
end proc:
seq(A340510(n), n=1..50) ; # R. J. Mathar, Jan 30 2024

Mathematica

a[n_] := a[n] = Switch[n, 1, 1, 2, 3, 3, 5, 4, 2, _, Module[{aa, ss, dd, an}, aa = Array[a, n-1]; ss = Sort[aa]; dd = Differences[ss]; For[an = Select[Transpose[{Rest[ss], dd}], #[[2]] == 1 &][[-1, 1]]+1, True, an++, If[FreeQ[aa = Array[a, n-1], an], If[Mod[Total[aa] + an, n+1] == 1, Return[an]]]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 31 2021 *)

Crossrefs

Cf. A000045, A001622, A019444.

Sequence in context: A282348 A026193 A026143 * A075626 A152649 A327263

Adjacent sequences: A340507 A340508 A340509 * A340511 A340512 A340513

Keyword

nonn,hear

Author

N. J. A. Sloane, Jan 28 2021

Extensions

More terms from Alois P. Heinz, Jan 28 2021

Status

approved