A175126 a(0) = a(1) = 0, for n >= 2, a(n) = number of steps of iteration of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = n.
0, 0, 1, 1, 2, 1, 3, 1, 4, 4, 5, 1, 6, 1, 7, 7, 8, 1, 9, 1, 10, 10, 11, 1, 12, 11, 13, 13, 14, 1, 15, 1, 16, 16, 17, 16, 18, 1, 19, 19, 20, 1, 21, 1, 22, 22, 23, 1, 24, 22, 25, 25, 26, 1, 27, 26, 28, 28, 29, 1, 30, 1, 31, 31, 32, 31, 33, 1, 34, 34, 35, 1, 36, 1, 37, 37, 38, 36, 39, 1, 40, 40
Offset: 0
Examples
Example (a(6)=3): 6-2=4, 4-2=2, 2-2=0; iterations has 3 steps. a(25) = 11, as we have 25 -> 20 -> 18 -> 16 -> 14 -> 12 -> 10 -> 8 -> 6 -> 4 -> 2 -> 0, in total eleven steps to reach zero. - _Antti Karttunen_, Aug 22 2019
Links
- Antti Karttunen, Table of n, a(n) for n = 0..16384
Programs
-
Maple
Contribution from R. J. Mathar, Mar 11 2010: (Start) A020639 := proc(n) min(op(numtheory[factorset](n))) ; end proc: A046666 := proc(n) n-A020639(n) ; end proc: A175126 := proc(n) local a; if n = 1 then 0; elif n = 0 then 0; else 1+procname(A046666(n)) ; end if; end proc: seq(A175126(n),n=1..100) ; (End)
-
Mathematica
stps[n_]:=Length[NestWhileList[#-FactorInteger[#][[1,1]]&,n,#>0&]]-1; Join[{0},Rest[Array[stps,90]]] (* Harvey P. Dale, Aug 15 2012 *) -
PARI
A020639(n) = if(1==n, n, factor(n)[1, 1]); A175126(n) = if(n<2,0,1+A175126(n-A020639(n))); \\ Antti Karttunen, Aug 22 2019
-
PARI
a(n) = if(n>1, (n-factor(n)[1, 1])/2 + 1, 0) \\ Jianing Song, Aug 07 2022
Formula
a(2n) = n >= 2; a(p) = 1 for p = prime.
a(n) = 0 if n<=1, else a(n) = 1+a(A046666(n)). - R. J. Mathar, Mar 11 2010
a(n) = (n-lpf(n))/2 + 1 for n > 1, lpf = A020639. - Jianing Song, Aug 07 2022
Extensions
Corrected A-number typo in the comment - R. J. Mathar, Mar 11 2010
Extended beyond a(30) by R. J. Mathar, Mar 11 2010
Term a(0) = 0 prepended by Antti Karttunen, Aug 22 2019
Comments