A274468 The length of the initial uninterrupted number of tau numbers in the chain defined by repeated subtraction of the number of divisors, starting with the n-th tau number.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 5, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Author

R. J. Mathar, Jun 24 2016

Keywords

• nonn

Comments

This is the persistence of the n-th tau number staying a tau number under the map x->A049820(x).

Records: 1, 2,...,8 occur at n=1, 6, 14, 16, 17, 7393, 7394, 8064,...

Examples

			a(196)=4 because the 196th tau number is 2016. Subtracting tau(2016)=36 gives 1980, which is a tau number. Subtracting tau(1980)=36 gives 1944, which is a tau number. Subtracting tau(1944)=24 gives 1920, which is a tau number. Subtracting tau(1920)=32 gives 1888 which is not a tau number. The length of the chain 2016->1980->1944->1920 is 4.
		

Links

• C. Meller, Tau numbers, June 2016.

Crossrefs

Cf. A033950, A049820.

Programs

Maple

isA033950 := proc(n)
    if n <= 0 then
        false;
    elif n = 1 then
        true;
    else
        modp(n, numtheory[tau](n)) = 0 ;
    end if;
end proc:
A274468 := proc(n)
    option remember;
    local a, t ;
    t := A033950(n) ;
    a := 1 ;
    while true do
        t := A049820(t) ;
        if isA033950(t) then
            a := a+1 ;
        else
            break;
        end if;
    end do:
    a ;
end proc:

Mathematica

isA033950[n_] := Which[n <= 0, False, n == 1, True, True, IntegerQ[ n/DivisorSigma[0, n]]];
A033950[n_] := A033950[n] = Module[{k}, If[n == 1, 1, For[k = A033950[n-1] + 1, True, k++, If[IntegerQ[k/DivisorSigma[0, k]], Return[k]]]]];
A274468[n_] := A274468[n] = Module[{a, t}, t = A033950[n]; a = 1; While[ True, t = t-DivisorSigma[0, t]; If[isA033950[t], a++, Break[]]]; a];
Table[A274468[n], {n, 1, 100}] (* Jean-François Alcover, Aug 11 2023, after R. J. Mathar *)