A105017 Positions of records in A064097.

1, 2, 3, 5, 7, 11, 19, 23, 43, 47, 94, 139, 235, 283, 517, 659, 1081, 1319, 2209, 2879, 5758, 8637, 13301, 20147, 30337, 49727, 61993, 103823, 135313, 247439, 366683, 606743, 811879, 1266767, 1739761, 2913671, 3797401, 5827343, 8288641, 16577282, 22784407, 37346483, 58003213, 81768767

Offset: 1

Author

Hugo Pfoertner, Feb 17 2006

Keywords

• nonn

Comments

With a(1) = 1, a(n) is the smallest number m such that the number of iterations of k -> k - k/p, p being any prime factor of k, needed to reach 1 starting at k = m is equal to n-1. (See Example section.) - Jaroslav Krizek, Feb 15 2010

a(n) =~ sqrt(e^(5n/6)). - Robert G. Wilson v, Aug 11 2022

Examples

			a(6)=11 because m=11 requires 6-1 = 5 iterations of r -> r - (largest divisor d < r) to reach 1 (the 5 iterations are 11-1=10, 10-5=5, 5-1=4, 4-2=2, and 2-1=1) and 11 is the smallest such number m. - _Jaroslav Krizek_, Feb 15 2010
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..200
• Hugo Pfoertner, Program

Crossrefs

Cf. A000079, A064097, A067513, A175125.

Programs

Maple

A105017 := proc()
    local maxa,a ;
    maxa := -999 ;
    for n from 1 do
        a := A064097(n) ;
        if a > maxa then
            printf("%d\n",n) ;
            maxa :=a ;
        end if;
    end do:
end proc:
A105017() ; # R. J. Mathar, Aug 07 2022

Mathematica

g[n_] := Block[{p = Select[1 + Divisors@n, PrimeQ]}, n*p/(p - 1)]; f[n_] := f[n] = Block[{lst = Union@Flatten[g@# & /@ f[n - 1]]}, If[ Length@ lst > 325, lst = Take[lst, 325 (* This limit must be increased for greater n's from the start. *) ]]; lst]; f[1] = {1}; f[0] = {0}; lst = {}; Do[ AppendTo[lst, Min[ f[n]]]; f[n - 1] =., {n, 44}]; lst (* Robert G. Wilson v, Aug 11 2022 *)

PARI

a=vectorsmall(10^7); a[1]=0;
for(n=2,#a,if(isprime(n),a[n]=1+a[n-1],f=factor(n);a[n]=a[f[1,1]]+a[n/f[1,1]])); \\ computes A064097
r=-oo; for(k=1,#a,if(a[k]>r,print1(k,", ");r=a[k])); \\ Hugo Pfoertner, Mar 16 2020

Extensions

a(1)=1 inserted by Robert G. Wilson v, Mar 16 2020