A358208 a(1) = 1; a(2) = 2; a(3) = 3; for n > 3, a(n) is the smallest positive number not occurring earlier that shares a factor with Sum_{k=1..n-1} A001065(k), where A001065(k) is the sum of the proper divisors of k.

1, 2, 3, 4, 5, 6, 8, 13, 10, 9, 12, 11, 7, 14, 15, 18, 16, 17, 20, 107, 21, 22, 24, 25, 191, 197, 27, 26, 28, 30, 33, 32, 35, 34, 36, 29, 38, 433, 39, 40, 42, 523, 577, 44, 45, 31, 677, 46, 48, 50, 23, 49, 52, 51, 54, 56, 55, 63, 43, 58, 37, 57, 53, 60, 66, 61, 62, 70, 68, 64, 65, 69, 71, 75, 80

Offset

1,2

Comments

The majority of terms are concentrated just below the line a(n) = n. However, some terms are much larger because the sum of the proper divisors of all previous terms is a prime number. In the first 10000 terms there are twenty-eight fixed points: 4, 5, 6, ..., 2486, 3280, 3292.

Conjecture: the sequence is a permutation of the positive integers.

Links

Table of n, a(n) for n=1..75.

Scott R. Shannon, Image of the first 10000 terms where a(n) is less than 110% of n. The green line is a(n) = n.

Example

a(7) = 8 as Sum_{k=1..6} A001065(k) = 0 + 1 + 1 + 3 + 1 + 6 = 12, and 8 is the smallest unused number that shares a factor with 12.

Crossrefs

Cf. A358209, A001065, A024916, A000203, A356851, A064413.

Sequence in context: A019532 A008537 A277195 * A133430 A033071 A306075

Adjacent sequences: A358205 A358206 A358207 * A358209 A358210 A358211

Keyword

nonn

Author

Scott R. Shannon, Nov 04 2022

Status

approved