Displaying 1-10 of 11 results found.
a(n) = the number of numbers k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments).
+10
8
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
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.
EXAMPLE
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.
PROG
(PARI)
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);
a(n) is the number of integers k <= n such that GCQ_A(n, k) >= 2 (see definition in comments).
+10
8
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
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 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.
EXAMPLE
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.
PROG
(PARI) 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])
(PARI) GCQ_A(a, b)=forstep(m=min(a, b)-1, 2, -1, if(a%m && b%m, return(m))); 0
a(n) = the sum of numbers k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments).
+10
8
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
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.
EXAMPLE
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.
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. A196440(n) = sum(k=1, n, (2<=GCQ_A(n, k))*k);
a(n) = the sum of numbers k <= n such that GCQ_A(n, k) >= 2 (see definition in comments).
+10
8
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
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.
EXAMPLE
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.
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
a(n) = the product of number k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments).
+10
8
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
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.
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.
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).
+10
8
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
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.
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.
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).
+10
7
1, 2, 4, 6, 12, 60, 60, 420
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.
EXAMPLE
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.
a(n) = the sum of GCQ_B(n, k) for 1 <= k <= n (see definition in comments).
+10
4
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
COMMENTS
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.
FORMULA
a(n) = n*(n-1) - 1 for n>= 5.
EXAMPLE
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.
CROSSREFS
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) = the sum of LCQ_B(n, k) for 1 <= k <= n (see definition in comments).
+10
4
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
COMMENTS
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.
FORMULA
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.
CROSSREFS
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).
a(n) = the sum of LCQ_A(n, k) for 1 <= k <= n (see definition in comments).
+10
3
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
COMMENTS
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.
EXAMPLE
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.
CROSSREFS
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).
Search completed in 0.011 seconds
| 