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

1,2

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

Links

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

Hugo Pfoertner, Program

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000079, A064097, A067513, A175125.

Sequence in context: A228062 A332216 A059878 * A214197 A083771 A158069

Adjacent sequences: A105014 A105015 A105016 * A105018 A105019 A105020

Keyword

nonn

Author

Hugo Pfoertner, Feb 17 2006

Extensions

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

Status

approved