A253630 Number of iterations of A253629 needed for n to reach 2.

0, 2, 1, 3, 2, 3, 2, 4, 3, 4, 3, 4, 3, 5, 3, 5, 4, 5, 4, 5, 4, 5, 4, 6, 4, 6, 4, 6, 5, 5, 4, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 7, 5, 6, 5, 6, 6, 7, 5, 7, 6, 7, 5, 7, 6, 7, 6, 6, 5, 7, 5, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 8, 6, 7, 6, 7, 6, 8, 6, 7, 6, 8, 6, 8, 6

Offset

2,2

Comments

If x or y is odd, then a(xy) = a(x) + a(y).

If x and y are both even, then a(xy) = a(x) + a(y) + 1.

Equivalently, if we define a function D by D(x) = a(x) if x is odd and D(x) = a(x) + 1 if x is even, then D is completely additive.

Links

Table of n, a(n) for n=2..88.

Colin Defant, An arithmetic function arising from the Dedekind psi function, arXiv:1501.00971 [math.NT], 2015.

Mathematica

L[n_] := If[EvenQ[n], (1/3) If[n > 1, n Times @@ (1 + 1/(Select[Divisors[n], PrimeQ])), 1], If[n > 1, n Times @@ (1 + 1/(Select[Divisors[n], PrimeQ])), 1]]; Table[Length@NestWhileList[L, n, # != 1 &] - 2, {n, 2, 260}]

Prog

(PARI) a253629(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2]-1)*if(f[i, 1]>2, f[i, 1]+1, 1)) ;
a(n) = my(nb = 0); my(m = n); while(m != 2, m = a253629(m); nb++); nb; \\ Michel Marcus, Jan 21 2015

Crossrefs

Cf. A253629.

Sequence in context: A070804 A303429 A303656 * A362938 A104481 A078709

Adjacent sequences: A253627 A253628 A253629 * A253631 A253632 A253633

Keyword

nonn

Author

Colin Defant, Jan 06 2015

Status

approved