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

0,5

Comments

See A005843 and A175127 for the smallest and greatest numbers m such that a(m) = k for k >= 2.

Links

Antti Karttunen, Table of n, a(n) for n = 0..16384

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A020639, A046666, A309892.

From a(2) on, one more than A046667.

Sequence in context: A361736 A278575 A333879 * A142884 A347028 A325365

Adjacent sequences: A175123 A175124 A175125 * A175127 A175128 A175129

Keyword

nonn,easy

Author

Jaroslav Krizek, Feb 15 2010

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

Status

approved