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A082091 a(n) = one more than the number of iterations of A005361 needed to reach 1 from the starting value n. 2
1, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 4, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 4, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4, 2, 2, 2, 3, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(1) = 1, and for n > 1, a(n) = 1 + a(A005361(n)).

EXAMPLE

For n = 2 = 2^1, A005361(2) = 1, so we reach 1 in one step, and thus a(2) = 1+1 = 2.

For n = 4 = 2^2, A005361(4) = 2; A005361(2) = 1, so we reach 1 in two steps, and thus a(4) = 2+1 = 3.

For n = 6 = 2^1 * 3^1, A005361(6) = 1*1 = 1, so we reach 1 in one step, and thus a(6) = 1+1 = 2.

For n = 64 = 2^6, A005361(64) = 6, thus a(64) = 1 + a(6) = 3.

For n = 10! = 3628800 = 2^8 * 3^4 * 5^2 * 7*1, A005361(3628800) = 64, thus a(3628800) = 1 + a(64) = 4.

MATHEMATICA

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] expr[x_] := Apply[Times, ep[x]] Table[Length[FixedPointList[expr, w]]-1, {w, 2, 128}]

(* Second program: *)

Table[Length@ NestWhileList[Apply[Times, FactorInteger[#][[All, -1]]] &, n, # != 1 &], {n, 105}] (* Michael De Vlieger, Jul 29 2017 *)

PROG

(PARI)

A005361(n) = factorback(factor(n)[, 2]); \\ This function from Charles R Greathouse IV, Nov 07 2014

A082091(n) = if(1==n, 1, 1+A082091(A005361(n))); \\ Antti Karttunen, Jul 28 2017

(PARI) first(n) = my(v = vector(n)); v[1] = 1; for(i=2, n, v[i] = v[factorback(factor(i)[, 2])] + 1); v \\ David A. Corneth, Jul 28 2017

(Scheme) (define (A082091 n) (if (= 1 n) n (+ 1 (A082091 (A005361 n))))) ;; Antti Karttunen, Jul 28 2017

CROSSREFS

Cf. A001414, A005361, A008475, A056239, A082083-A082086, A082090.

Sequence in context: A098397 A278744 A346213 * A334216 A100549 A085962

Adjacent sequences:  A082088 A082089 A082090 * A082092 A082093 A082094

KEYWORD

nonn

AUTHOR

Labos Elemer, Apr 09 2003

EXTENSIONS

Term a(1)=1 prepended, Name and Example sections edited by Antti Karttunen, Jul 28 2017

STATUS

approved

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Last modified July 30 23:06 EDT 2021. Contains 346365 sequences. (Running on oeis4.)