A196442 a(1) = a(2) = 1; for n >= 2, a(n) is the product of number k <= n such that GCQ_A(n, k) >= 2 (see definition in comments).

1, 1, 3, 4, 60, 30, 2520, 6720, 45360, 604800, 19958400, 3991680, 3113510400, 14529715200, 163459296000, 3487131648000, 177843714048000, 266765571072000, 60822550204416000, 67580611338240000, 6386367771463680000, 187333454629601280000, 12926008369442488320000, 5170403347776995328000, 7755605021665492992000000, 67215243521100939264000000

Offset

1,3

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.

a(n) is also the sum of number k <= n such that LCQ_A(n, k) >= 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

Table of n, a(n) for n=1..26.

Formula

a(n) = A000142(n) / A196441(n).

Example

For n = 6, a(6) = 30 because there are 2 cases k (k = 5, 6) with GCQ_A(6, k) >= 2: GCQ_A(6, 5) = 4, GCQ_A(6, 6) = 5 and the product of these numbers k is 30.
Also there are 2 same cases k with LCQ_A(6, k) >= 2: 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.
A196442(n) = prod(k=1, n, if(2<=GCQ_A(n, k), k, 1)); \\ Antti Karttunen, Jun 13 2018

Crossrefs

Cf. A196437, A196438, A196439, A196440, A196441, A196442, A196443, A196444.

Sequence in context: A331725 A067093 A041105 * A349587 A278035 A079076

Adjacent sequences: A196439 A196440 A196441 * A196443 A196444 A196445

Keyword

nonn

Author

Jaroslav Krizek, Nov 26 2011

Extensions

More terms from Antti Karttunen, Jun 13 2018

Status

approved