A168007 Jumping divisor sequence (see Comments lines for definition).

1, 2, 4, 3, 6, 5, 10, 9, 12, 11, 22, 21, 24, 23, 46, 45, 48, 47, 94, 93, 96, 95, 100, 99, 102, 101, 202, 201, 204, 203, 210, 209, 220, 219, 222, 221, 234, 233, 466, 465, 468, 467, 934, 933, 936, 935, 940, 939, 942, 941, 1882, 1881, 1884, 1883, 1890, 1889, 3778, 3777, 3780, 3779, 7558, 7557, 7560, 7559

Offset: 1

Author

Omar E. Pol, Nov 19 2009

Keywords

• nonn
• easy

Comments

Consider the diagram with overlapping periodic curves that appears in the Links section (figure 2). The number of curves that contain the point [n,0] equals the number of divisors of n. The curve of diameter d represents the divisor d of n. Now consider only the lower part of the diagram (figure 3). Starting from point [1,0] we continue our journey walking along the semicircumference with smallest diameter not used previously (see the illustration of initial terms, figure 1). The sequence is formed by the values of n where the trajectory intercepts the x axis. - Omar E. Pol, Jan 14 2019

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1000
• Omar E. Pol, Illustration of initial terms (Fig. 1)
• Omar E. Pol, Periodic curves and tau(n) (Fig. 2)
• Omar E. Pol, Periodic curves and tau(n), lower part upside down (Fig. 3)

Crossrefs

Cf. A000005, A002808, A020639, A004280, A168008, A168009.

Programs

PARI

lista(nn) = {my(v=vector(nn, i, if(i<4, 2^i/2))); for(n=4, nn, if(v[n-1]%2, v[n]=v[n-1] + factor(v[n-1])[1, 1], v[n]=v[n-1] - 1)); v; } \\ Jinyuan Wang, Mar 14 2020

Python

from itertools import count, islice
from sympy import primefactors
def A168007_gen(): # generator of terms
    yield (a := 1)
    for n in count(2):
        yield (a:=a+(min(primefactors(a),default=1) if a&1 or a==2 else -1))
A168007_list = list(islice(A168007_gen(),20)) # Chai Wah Wu, Mar 14 2023

Formula

a(1) = 1; if a(n) is an even composite number then a(n+1) = a(n) - 1; otherwise a(n+1) = a(n) + A020639(a(n)). - Omar E. Pol, Jan 13 2019

Extensions

More terms from Omar E. Pol, Jan 12 2019