A381466 a(0) = 4; for n > 0, a(n) = a(n-1) + n if G = 1 or a(n) = n/G if G > 1, where G = gcd(a(n-1), n).

4, 5, 7, 10, 2, 7, 13, 20, 2, 11, 21, 32, 3, 16, 7, 22, 8, 25, 43, 62, 10, 31, 53, 76, 6, 31, 57, 9, 37, 66, 5, 36, 8, 41, 75, 7, 43, 80, 19, 58, 20, 61, 103, 146, 22, 67, 113, 160, 3, 52, 25, 76, 13, 66, 9, 64, 7, 64, 29, 88, 15, 76, 31, 94, 32, 97, 163, 230, 34, 103, 173, 244

Offset: 0

Author

Sam Chapman, Feb 24 2025

Keywords

• nonn

Comments

If a(n) < n for some n, then a(n+1) > n+1.

If a(n) > n, a(n+1) > n+1, and a(n+2) > n+2 for some n, then a(n+3) < n+3.

1 < a(n) for all n.

sqrt(n/6) < a(n) <= 7n/2 - 9/2 for all n.

a(p)>p for all primes p.

If one were to use the same rule to generate this sequence with any other initial value that is congruent to 4 or 8 (mod 12), that sequence would agree with this one for all n>3.

If one were to use the same rule to generate this sequence with an initial term that is not congruent to 4 or 8 (mod 12), then it would output the number 1 before the 5th term. When a sequence follows a(n)’s rules and outputs the number 1 at some index k, one gets the following quasi-periodic behavior: 1, k+2, 1, k+4, 1, k+6, etc., and are as such “boring” sequences.

Examples

			a(12) = 3 and gcd(3, 13) = 1, so a(13) = 3 + 13 = 16.  gcd(16, 14) = 2, so a(14) = 14/2 = 7.
		

Links

• Sam Chapman, Table of n, a(n) for n = 0..10000

Crossrefs

Similar to A133058, A091508.

Programs

Mathematica

s={4};Do[G=GCD[s[[-1]],n];AppendTo[s,If[G==1,s[[-1]]+n,n/G]],{n,71}];s (* James C. McMahon, Mar 02 2025 *)

PARI

lista(nn) = my(v = vector(nn)); v[1] = 4; for (n=2, nn, my(g=gcd(v[n-1], n-1)); if (g==1, v[n] = v[n-1] + n-1, v[n] = (n-1)/g);); v; \\ Michel Marcus, Feb 26 2025