A394694 Number of occurrences of n in A081493.

1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 4, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 5, 2, 2, 1, 6, 1, 2, 3, 4, 2, 4, 1, 3, 2, 4, 1, 7, 1, 2, 3, 3, 2, 4, 1, 5, 3, 2, 1, 6, 2, 2, 2, 4

Offset

1,3

Comments

Conjecturally: a(n) = A000005(n-1)/2 + e(n), where e(n) = 1/2 if n-1 is a perfect square, 1 if n-1 is pronic, and 0 otherwise.

Here A000005(k) is the number of divisors of k.

In particular, a(n) = 1 for all n of the form p+1 where p is an odd prime (since A000005(p) = 2). Example: a(4) = 1 since 4 = 3 + 1 and 3 is prime.

Also, the number of occurrences of n in A209297 is A033831(n-1) for n >= 2.

Also conjecturally: a(n) + A033831(n-1) = A000005(n-1) for n >= 2.

Equivalently, A033831(n-1) = A000005(n-1)/2 + d(n), where d(n) = -1/2 if n-1 is a perfect square, -1 if n-1 is pronic, and 0 otherwise.

Links

Table of n, a(n) for n=1..89.

Example

Initial rows of A081493:
  1;
  2,  3;
  3,  5,  7;
  4,  7, 10, 13;
  5,  9, 13, 17, 21;
  6, 11, 16, 21, 26, 31;
  7, 13, 19, 25, 31, 37, 43.
The following counts are obtained directly from the above rows:
n = 5: 5 occurs twice, so a(5) = 2. Here n-1 = 4 (square), tau(4) = 3, and (3+1)/2 = 2.
n = 7: 7 occurs three times, so a(7) = 3. Here n-1 = 6 (pronic), tau(6) = 4, and (4+2)/2 = 3.
n = 6: 6 occurs once, so a(6) = 1. Here n-1 = 5, tau(5) = 2, and 2/2 = 1.

Mathematica

a[n_]:=Count[Table[NestList[#+i-1&, i, i-1], {i, n}]//Flatten, n]; Array[a, 89] (* James C. McMahon, Apr 08 2026 *)

Prog

(Python)
from collections import defaultdict
def a_list(N):
    freq = defaultdict(int)
    for n in range(1, N+1):
        for k in range(n):
            val = n + k*(n-1)
            if val <= N:
                freq[val] += 1
    return [freq[n] for n in range(1, N+1)]
print(a_list(90))

Crossrefs

Cf. A081493, A033831, A000005, A209297.

Sequence in context: A318831 A303710 A263280 * A136107 A178691 A329313

Adjacent sequences: A394691 A394692 A394693 * A394695 A394696 A394697

Keyword

nonn,easy

Author

Gokul Padmanabhan, Mar 29 2026

Status

approved