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A360464 a(n) = a(n-1) + a(n-2) - a(n-3) + gcd(a(n-1), a(n-3)), with a(1) = a(2) = a(3) = 1. 2
1, 1, 1, 2, 3, 5, 7, 10, 17, 21, 29, 34, 43, 49, 59, 66, 77, 85, 97, 106, 119, 129, 143, 154, 169, 193, 209, 234, 251, 277, 295, 322, 341, 369, 389, 418, 439, 469, 491, 522, 545, 577, 601, 634, 659, 693, 719, 754, 781, 817, 845, 882, 911, 949, 979, 1018, 1049 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
All terms beyond a(458) are divisible by 5. - Jack Braxton, Feb 14 2023
From Robert Israel, Feb 15 2023: (Start)
a(n) is divisible by 25 for n >= 8857.
a(n) is divisible by 125 for n >= 8861.
a(n) is divisible by 625 for n >= 8945.
a(n) is divisible by 1875 for n >= 9060.
a(n) is divisible by 5625 for n >= 9064.
Do there exist N > 9064 and m > 5625 such that a(n) is divisible by m for n >= N? If so, N >= 2*10^7. (End)
From Pontus von Brömssen, Feb 17 2023: (Start)
(Answer to the question above.) Yes:
a(n) has an additional factor 5 for n >= 64423404 (so a(n) is divisible by 28125);
a(n) has an additional factor 5 for n >= 64423410;
a(n) has an additional factor 3 for n >= 64424073;
a(n) has an additional factor 21 for n >= 64424144;
a(n) has an additional factor 3 for n >= 64428745;
a(n) has an additional factor 7 for n >= 64428748;
a(n) has an additional factor 3 for n >= 64428756;
a(n) has an additional factor 3 for n >= 64428821;
a(n) has an additional factor 3 for n >= 64514757;
a(n) has an additional factor 5 for n >= 64514783;
a(n) has an additional factor 3 for n >= 797299454;
a(n) has an additional factor 3 for n >= 797299480;
a(n) has an additional factor 5 for n >= 797299487;
a(n) has an additional factor 3 for n >= 797299490;
a(n) has an additional factor 5 for n >= 797299652;
a(n) has an additional factor 3 for n >= 797299667;
a(n) has an additional factor 7 for n >= 797299846;
a(n) has an additional factor 3 for n >= 797299933.
The index for which the next additional factor occurs (if it exists) is larger than 2*10^10.
(End)
LINKS
FORMULA
a(n) = a(n-1) + a(n-2) - a(n-3) + gcd(a(n-1), a(n-3)).
EXAMPLE
a(5) = 2 + 1 - 1 + gcd(2, 1) = 3.
MAPLE
A:= Vector(200):
A[1]:= 1: A[2]:= 1: A[3]:= 1:
for n from 4 to 200 do
A[n]:= A[n-1] + A[n-2] - A[n-3] + igcd(A[n-1], A[n-3])
od:
convert(A, list); # Robert Israel, Feb 15 2023
MATHEMATICA
a[1] = a[2] = a[3] = 1; a[n_] := a[n] = a[n-1] + a[n-2] - a[n-3] + GCD[a[n-1], a[n-3]]; Array[a, 100] (* Amiram Eldar, Feb 08 2023 *)
PROG
(Python)
from math import gcd
a = [0, 1, 1, 1]
[a.append(a[n-1]+a[n-2]-a[n-3]+gcd(a[n-1], a[n-3])) for n in range(4, 58)]
print(a[1:]) # Michael S. Branicky, Feb 09 2023
CROSSREFS
Sequence in context: A052011 A278706 A005468 * A271063 A182521 A283809
KEYWORD
nonn,easy
AUTHOR
Jack Braxton, Feb 08 2023
STATUS
approved

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Last modified June 20 19:46 EDT 2024. Contains 373532 sequences. (Running on oeis4.)