A196441 a(n) = the product of number k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments).

1, 2, 2, 6, 2, 24, 2, 6, 8, 6, 2, 120, 2, 6, 8, 6, 2, 24, 2, 36, 8, 6, 2, 120, 2, 6, 8, 6, 2, 24, 2, 6, 8, 6, 2, 120, 2, 6, 8, 36, 2, 24, 2, 6, 8, 6, 2, 120, 2, 6, 8, 6, 2, 24, 2, 6, 8, 6, 2, 5040, 2, 6, 8, 6, 2, 24, 2, 6, 8, 6, 2, 120, 2, 6, 8, 6, 2, 24, 2, 36, 8, 6, 2, 120, 2, 6, 8, 6, 2, 24, 2, 6, 8, 6, 2, 120, 2, 6, 8, 36, 2, 24, 2, 6, 8

Offset

1,2

Comments

Definition of GCQ_A: The greatest common non-divisor of type A (GCQ_A) of two positive integers a and b (a<=b) is the largest positive non-divisor q of numbers a and b such that 1<=q<=a common to a and b; GCQ_A(a, b) = 0 if no such c exists.

GCQ_A(1, b) = GCQ_A(2, b) = 0 for b >=1. GCQ_A(a, b) = 0 or >= 2.

Definition of LCQ_A: The least common non-divisor of type A (LCQ_A) of two positive integers a and b (a<=b) is the least positive non-divisor q of numbers a and b such that 1<=q<=a common to a and b; LCQ_A(a, b) = 0 if no such c exists. LCQ_A(1, b) = LCQ_A(2, b) = 0 for b >=1. LCQ_A(a, b) = 0 or >= 2.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = A000142(n) / A196442(n).

Example

For n = 6, a(6) = 24 because there are 4 cases k (k = 1, 2, 3, 4) with GCQ_A(6, k) = 0:
GCQ_A(6, 1) = 0, GCQ_A(6, 2) = 0, GCQ_A(6, 3) = 0, GCQ_A(6, 4) = 0, GCQ_A(6, 5) = 4, GCQ_A(6, 6) = 5. Product of such numbers k is 24.
Also there are 4 same cases k with LCQ_A(6, k) = 0:
LCQ_A(6, 1) = 0, LCQ_A(6, 2) = 0, LCQ_A(6, 3) = 0, LCQ_A(6, 4) = 0, LCQ_A(6, 5) = 4, LCQ_A(6, 6) = 4.

Prog

(PARI)
GCQ_A(a, b) = { forstep(m=min(a, b)-1, 2, -1, if(a%m && b%m, return(m))); 0; }; \\ From PARI-program in A196438.
A196441(n) = prod(k=1, n, if(2<=GCQ_A(n, k), 1, k)); \\ Antti Karttunen, Jun 13 2018

Crossrefs

Cf. A196437, A196438, A196439, A196440, A196442, A196443, A196444.

Sequence in context: A013608 A369867 A324158 * A130674 A284839 A286376

Adjacent sequences: A196438 A196439 A196440 * A196442 A196443 A196444

Keyword

nonn

Author

Jaroslav Krizek, Nov 26 2011

Extensions

More terms from Antti Karttunen, Jun 13 2018

Status

approved