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

Author

N. J. A. Sloane, Jan 28 2021

Keywords

• nonn
• hear

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.

Crossrefs

Cf. A000045, A001622, A019444.

Programs

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 *)

Formula

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

Extensions

More terms from Alois P. Heinz, Jan 28 2021