A196443 -id:A196443 - cp's OEIS frontend

A196437 a(n) = the number of numbers k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments).

1, 2, 2, 3, 2, 4, 2, 3, 3, 3, 2, 5, 2, 3, 3, 3, 2, 4, 2, 4, 3, 3, 2, 5, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 5, 2, 3, 3, 4, 2, 4, 2, 3, 3, 3, 2, 5, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 7, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 5, 2, 3, 3, 3, 2, 4, 2, 4, 3, 3, 2, 5, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 5, 2, 3, 3, 4, 2, 4, 2, 3, 3

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

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.

			For n = 6, a(6) = 4 because there are 4 cases 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.
Also there are 4 cases 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.

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

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

GCQ_A(a, b) = { forstep(m=min(a, b)-1, 2, -1, if(a%m && b%m, return(m))); 0; };
A196438(n) = sum(i=3, n, GCQ_A(i, n)>=2);
A196437(n) = (n - A196438(n)); \\ Antti Karttunen, Mar 20 2018, based on Charles R Greathouse IV's Aug 26 2017 PARI-program in A196438.

a(n) = n - A196438(n).

More terms from Antti Karttunen, Mar 20 2018

A196438 a(n) is the number of integers k <= n such that GCQ_A(n, k) >= 2 (see definition in comments).

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 5, 5, 6, 7, 9, 7, 11, 11, 12, 13, 15, 14, 17, 16, 18, 19, 21, 19, 23, 23, 24, 25, 27, 26, 29, 29, 30, 31, 33, 31, 35, 35, 36, 36, 39, 38, 41, 41, 42, 43, 45, 43, 47, 47

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

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 number 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.

			For n = 6, a(6) = 2 because there are 2 cases with GCQ_A(6, k) >= 2:
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.
Also there are 2 cases with LCQ_A(6, k) >= 2:
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.

David A. Corneth, Table of n, a(n) for n = 1..10000

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

GCQ_A(a, b)=m = min(a, b); if(m < 3, return(0)); da = Set(divisors(a)); db = Set(divisors(b)); s = Set(vector(m-1,i,i)); s = setminus(s, da); s = setminus(s, db); if(#s==0,0,s[#s])
a(n) = sum(i=3,n,GCQ_A(i, n)>=2) \\ David A. Corneth, Aug 04 2017

GCQ_A(a, b)=forstep(m=min(a,b)-1,2,-1, if(a%m && b%m, return(m))); 0
a(n) = sum(i=3,n,GCQ_A(i, n)>=2) \\ Charles R Greathouse IV, Aug 26 2017

a(n) = n - A196437(n).

A196439 a(n) = the sum of numbers k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments).

Original entry on oeis.org

1, 3, 3, 6, 3, 10, 3, 6, 7, 6, 3, 15, 3, 6, 7, 6, 3, 10, 3, 12, 7, 6, 3, 15, 3, 6, 7, 6, 3, 10, 3, 6, 7, 6, 3, 15, 3, 6, 7, 12, 3, 10, 3, 6, 7, 6, 3, 15, 3, 6, 7, 6, 3, 10, 3, 6, 7, 6, 3, 28, 3, 6, 7, 6, 3, 10, 3, 6, 7, 6, 3, 15, 3, 6, 7, 6, 3, 10, 3, 12, 7, 6, 3, 15, 3, 6, 7, 6, 3, 10, 3, 6, 7, 6, 3, 15, 3, 6, 7, 12, 3, 10, 3, 6, 7

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

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.

			For n = 6, a(6) = 10 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. Sum of such numbers k is 10.
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.

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

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

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.
A196440(n) = sum(k=1,n,(2<=GCQ_A(n,k))*k);
A196439(n) = (((n*(n+1))/2) - A196440(n)); \\ Antti Karttunen, Jun 12 2018

a(n) = A000217(n) - A196440(n).

More terms from Antti Karttunen, Jun 12 2018

A196440 a(n) = the sum of numbers k <= n such that GCQ_A(n, k) >= 2 (see definition in comments).

Original entry on oeis.org

0, 0, 3, 4, 12, 11, 25, 30, 38, 49, 63, 63, 88, 99, 113, 130, 150, 161, 187, 198, 224, 247, 273, 285, 322, 345, 371, 400, 432, 455, 493, 522, 554, 589, 627, 651, 700, 735, 773, 808, 858, 893, 943, 984, 1028, 1075, 1125, 1161, 1222, 1269, 1319, 1372, 1428, 1475, 1537, 1590, 1646, 1705, 1767, 1802, 1888, 1947, 2009, 2074, 2142, 2201, 2275

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

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.

			For n = 6, a(6) = 11 because there are 2 cases k (k = 5, 6) with GCQ_A(6, k) >= 2:
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. Sum of such numbers k is 11.
Also there are 2 same cases k  with LCQ_A(6, k) >= 2:
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.

Antti Karttunen, Table of n, a(n) for n = 1..1001

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

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
A196440(n) = sum(k=1,n,(2<=GCQ_A(n,k))*k); \\ Antti Karttunen, Jun 12 2018

a(n) = A000217(n) - A196439(n).

More terms from Antti Karttunen, Jun 12 2018

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

Original entry on oeis.org

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

Jaroslav Krizek, Nov 26 2011

nonn

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.

			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.

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

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

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

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

More terms from Antti Karttunen, Jun 13 2018

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).

Original entry on oeis.org

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

Jaroslav Krizek, Nov 26 2011

nonn

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.

			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.

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

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

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

More terms from Antti Karttunen, Jun 13 2018

A196444 a(n) = the smallest number m such that GCQ_A(m, k) = LCQ_A(m, k) = 0 for all 1 <= k <= n (see definition in comments).

Original entry on oeis.org

1, 2, 4, 6, 12, 60, 60, 420

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

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.

			For a(5) = 12 holds: GCQ_A(12, 1) = GCQ_A(12, 2) = GCQ_A(12, 3) = GCQ_A(12, 4) = GCQ_A(12, 5) = 0.
Also holds: LCQ_A(12, 1) = LCQ_A(12, 2) = LCQ_A(12, 3) = LCQ_A(12, 4) = LCQ_A(12, 5) = 0.

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

A199972 a(n) = the sum of GCQ_B(n, k) for 1 <= k <= n (see definition in comments).

Original entry on oeis.org

0, 0, 4, 9, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, 1405, 1481, 1559, 1639, 1721, 1805, 1891, 1979, 2069

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

Definition of GCQ_B: The greatest common non-divisor of type B (GCQ_B) 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<=b common to a and b; GCQ_B(a, b) = 0 if no such c exists.

For b>=5 holds: GCQ_B(a, b) = b - 1 if a = b or a<= b-2, GCQ_B(a, b) = b - 2 if a = b-1.

			For n = 4, a(4) = 9 because GCQ_B(4, 1) = 3, GCQ_B(4, 2) = 3, GCQ_B(4, 3) = 0, GCQ_B(4, 4) = 3 and sum of results is 9.
For n = 5, a(4) = 19 because GCQ_B(5, 1) = 4, GCQ_B(5, 2) = 4, GCQ_B(5, 3) = 4, GCQ_B(5, 4) = 3, GCQ_B(5, 5) = 4 and sum of results is 19.

Cf.: A196443 (the sum of GCQ_A(n, k) for 1 <= k <= n).
Cf.: A199973 (the sum of LCQ_B(n, k) for 1 <= k <= n).
Cf.: A199971 (the sum of LCQ_A(n, k) for 1 <= k <= n).
Cf.: A199973 (the sum of LCQ_C(n, k) for 1 <= k <= n).

a(n) = n*(n-1) - 1 for n>= 5.

A199973 a(n) = the sum of LCQ_B(n, k) for 1 <= k <= n (see definition in comments).

Original entry on oeis.org

0, 0, 4, 9, 12, 25, 18, 28, 28, 33, 28, 64, 35, 47, 51, 59, 45, 76, 51, 81, 68, 74, 61, 128, 72, 88, 87, 103, 78, 145, 84, 119, 107, 114, 101, 195, 101, 129, 126, 166

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

Definition of LCQ_B: The least common non-divisor of type B (LCQ_B) 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<=b common to a and b; LCQ_B(a, b) = 0 if no such c exists.

LCQ_B(a, b) = 0 or >= 2.

Cf.: A196443 (the sum of GCQ_A(n, k) for 1 <= k <= n).
Cf.: A199972 (the sum of GCQ_B(n, k) for 1 <= k <= n).
Cf.: A199971 (the sum of LCQ_A(n, k) for 1 <= k <= n).
Cf.: A199974 (the sum of LCQ_C(n, k) for 1 <= k <= n).

For n = 6, a(6) = 9 because LCQ_B(6, 1) = 4, LCQ_B(6, 2) = 4, LCQ_B(6, 3) = 4, LCQ_B(6, 4) = 5, LCQ_B(6, 5) = 4, LCQ_B(6, 6) = 4. Sum of results is 25.

A199971 a(n) = the sum of LCQ_A(n, k) for 1 <= k <= n (see definition in comments).

Original entry on oeis.org

0, 0, 2, 3, 7, 8, 13, 17, 17, 23, 23, 37, 30, 37, 39, 48, 40, 59, 46, 62, 57, 64, 56, 101, 67, 78, 76, 92, 73, 126, 79, 108, 96, 104, 96, 168, 96, 119, 115, 147

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

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.

			For n = 6, a(6) = 9 because 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. Sum of results is 8.

Cf.: A196443 (the sum of GCQ_A(n, k) for 1 <= k <= n).
Cf.: A199972 (the sum of GCQ_B(n, k) for 1 <= k <= n).
Cf.: A199973 (the sum of LCQ_B(n, k) for 1 <= k <= n).
Cf.: A199974 (the sum of LCQ_C(n, k) for 1 <= k <= n).

cp's OEIS Frontend

A196437 a(n) = the number of numbers k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments).

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A196438 a(n) is the number of integers k <= n such that GCQ_A(n, k) >= 2 (see definition in comments).

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A196439 a(n) = the sum of numbers k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments).

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A196440 a(n) = the sum of numbers k <= n such that GCQ_A(n, k) >= 2 (see definition in comments).

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

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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).

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A196444 a(n) = the smallest number m such that GCQ_A(m, k) = LCQ_A(m, k) = 0 for all 1 <= k <= n (see definition in comments).

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A199972 a(n) = the sum of GCQ_B(n, k) for 1 <= k <= n (see definition in comments).