A364189 a(n) is the smallest k such that A000005(j) = A000005(j-m) for j = k..k+n-1 for some m > 0.

3, 7, 17, 31, 71, 232, 412, 1756, 2759, 3763, 4881, 4881, 26812, 125804, 566658, 566658, 1601927, 1601927, 18641185, 42401324, 131296837, 136407785

Offset

1,1

Comments

Equivalently, a(n) is the smallest index at which there begins an n-term subsequence of A000005 that has occurred before.

Links

Table of n, a(n) for n=1..22.

Example

Let d(j) = A000005(j) (i.e., the number of divisors of j). The sequence {d(j)} = A000005 begins as follows:
.
     j: 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 ...
  d(j): 1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6 ...
.
The first value of d(j) that appears more than once is 2, which appears for the first time at j=2 and for the second time at j=3. Thus a(1) = 3.
Similarly, of all the subsequences of {d(j)} consisting of two consecutive terms, the first one to appear more than once is {2,4}, which first appears beginning at j=5 and then appears the second time beginning at j=7. Thus a(2) = 7.
Likewise, the first subsequence of {d(j)} consisting of three consecutive terms to appear a second time is {2,6,2}, which first appears beginning at j=11 and then reappears beginning at j=17, so a(3) = 17.
The first repeated 1-, 2-, and 3-term subsequences and the corresponding terms a(1), a(2), and a(3) are illustrated below.
.
            a(1)        a(2)                          a(3)
              |           |                             |
     j: 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
  d(j): 1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
           ^  ^     ^--^  ^--^        ^--^--^           ^--^--^

Mathematica

nn = 25000; s = {}; k = 1; f[x_] := DivisorSigma[0, x]; Monitor[Do[AppendTo[s, f[n]]; t = SequencePosition[s[[1 ;; n - k]], s[[-k ;; -1]]]; If[Length[t] > 0, k++; a[k-1] = n - k + 2], {n, nn}], n]; TakeWhile[Array[a, k], IntegerQ] (* Michael De Vlieger, Aug 07 2023 *)

Prog

(Python)
from itertools import count
from sympy import divisor_count
def A364189(n):
    b = {a:=tuple(divisor_count(i) for i in range(1, n+1)), }
    for m in count(n+1):
        if (a:=a[1:]+(divisor_count(m), )) in b:
            return m-n+1
        b.add(a) # Chai Wah Wu, Aug 07 2023

Crossrefs

Cf. A000005.

Sequence in context: A068682 A292446 A045425 * A099983 A275631 A048860

Adjacent sequences: A364186 A364187 A364188 * A364190 A364191 A364192

Keyword

nonn,more

Author

Jon E. Schoenfield, Aug 07 2023

Extensions

a(14) from Michael De Vlieger, Aug 07 2023

a(15)-a(22) from Chai Wah Wu, Aug 07 2023

Status

approved