cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A196443 a(n) = the sum of GCQ_A(n, k) for 1 <= k <= n (see definition in comments).

Original entry on oeis.org

0, 0, 2, 3, 9, 9, 20, 24, 32, 41, 54, 55, 77, 87, 100, 115, 135, 145, 170, 180, 205, 227, 252, 263, 298, 321, 346, 372, 405, 424, 464, 490, 523, 557, 592, 616, 665, 699, 736, 768, 819, 850, 902, 940, 983, 1031, 1080, 1113, 1174, 1219
Offset: 1

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Author

Jaroslav Krizek, Nov 26 2011

Keywords

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.

Examples

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

Crossrefs

Cf. A196437, A196438, A196439, A196440, A196441, A196442, A196444, A018804.
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).

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

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Author

Jaroslav Krizek, Nov 26 2011

Keywords

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.

Examples

			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).

Formula

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

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

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Author

Jaroslav Krizek, Nov 26 2011

Keywords

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.

Examples

			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).

A199974 a(n) = the sum of LCQ_C(n, k) for 1 <= k <= n (see definition in comments).

Original entry on oeis.org

2, 6, 7, 14, 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

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Author

Jaroslav Krizek, Nov 26 2011

Keywords

Comments

Definition of LCQ_C: The least common non-divisor of type C (LCQ_C) 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 common to a and b. LCQ_C(a, b) >= 2.

Examples

			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.: A199973 (the sum of LCQ_B(n, k) for 1 <= k <= n).
Showing 1-4 of 4 results.

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