A364934 a(n+1) = 1 + number of previous terms that share a factor > 1 with a(n); a(1) = 2.

2, 2, 3, 2, 4, 5, 2, 6, 8, 8, 9, 4, 10, 12, 14, 13, 2, 14, 15, 8, 16, 17, 2, 18, 22, 20, 23, 2, 22, 23, 3, 8, 24, 29, 2, 26, 28, 28, 29, 3, 10, 32, 31, 2, 32, 33, 13, 4, 34, 36, 42, 43, 2, 38, 39, 17, 4, 40, 43, 3, 15, 21, 21, 22, 43, 4, 43, 5, 9, 19, 3, 20, 48, 58, 48, 60, 63, 28, 52, 53, 2, 51, 28

Offset: 1

Author

Rok Cestnik, Aug 15 2023

Keywords

• nonn

Comments

There are prominent lines that have more terms, their coefficients are approximately: 0.519, 0.329, 0.689, 0.188, 0.615, ... (see the frequency link). They seem to be distorted prime harmonic lines: 1/2, 1/3, 2/3, 1/5, 3/5, ... from A016035.

It appears limsup a(n)/n is approximately 0.83.

Examples

			[2,*]          1 term  shares a factor with 2, so a(2) = 1+1 = 2.
[2,2,*]        2 terms share  a factor with 2, so a(3) = 1+2 = 3.
[2,2,3,*]      1 term  shares a factor with 3, so a(4) = 1+1 = 2.
[2,2,3,2,*]    3 terms share  a factor with 2, so a(5) = 1+3 = 4.
[2,2,3,2,4,*]  4 terms share  a factor with 4, so a(6) = 1+4 = 5.
		

Links

• Rok Cestnik, Table of n, a(n) for n = 1..1000
• Rok Cestnik, a(n)/n frequency for 3*10^5 terms.
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Crossrefs

Cf. A016035, A124056.

Programs

Mathematica

nn = 120; c[] := 0; s = {}; a[1] = j = 2; Do[c[j]++; If[c[j] == 1, AppendTo[s, j]]; k = 1 + Total@ Map[c[#] Boole[GCD[#, j] > 1] &, s]; Set[{a[n], j}, {k, k}], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Aug 16 2023 *)

Python

from sympy.ntheory import primefactors
a=[2]; p = [{2}];
for n in range(1,1000):
    a.append(sum(1 if p[-1].intersection(p[i]) else 0 for i in range(n))+1)
    p.append(set(primefactors(a[-1])))

Python

from math import gcd, prod
from sympy import factorint
from itertools import islice
from collections import Counter
def agen(): # generator of terms
    a=[2]; f=2; c=Counter([f])
    while True:
        yield a[-1]
        a.append(1 + sum(c[fi] for fi in c if gcd(f,fi)>1))
        f=prod(factorint(a[-1]).keys())
        c[f] += 1
a=list(islice(agen(), 1000)) # Michael S. Branicky, Aug 16 2023