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.

User: Shanzhen Gao

Shanzhen Gao's wiki page.

Shanzhen Gao has authored 51 sequences. Here are the ten most recent ones:

A227212 Generalized Markoff numbers: largest of 7-tuple of positive numbers a, b, c, d, e, f, g satisfying the Markoff(7) equation a^2+b^2+c^2+d^2+e^2+f^2+g^2 = 5abcdefg.

Original entry on oeis.org

2, 3, 9, 14, 43, 67, 89, 206, 209, 321, 881, 987, 1538, 1934, 3121, 4003, 4689, 4729, 7369, 8009, 8721, 14627, 22658, 35307, 43889, 44289, 46606, 86329, 86987
Offset: 1

Views

Author

Shanzhen Gao, Sep 19 2013

Keywords

Examples

			a(1)=2 and a(2)=3 since (2, 1, 1, 1, 1, 1, 1) and (3, 1, 1, 1, 1, 1, 1) satisfying a^2+b^2+c^2+d^2+e^2+f^2+g^2=5abcdefg.

Crossrefs

Cf. A227210, A227211, A227214.

A227214 Generalized Markoff numbers: largest of 7-tuple of positive numbers a, b, c, d, e, f, g satisfying the Markoff(7) equation a^2+b^2+c^2+d^2+e^2+f^2+g^2 = 7abcdefg.

Original entry on oeis.org

1, 6, 41, 281, 1721, 1926, 13201, 72241, 80646, 90481, 493921, 620166, 2963561, 3032401, 3788441, 4250681, 23145121, 29134601, 127288601, 141753606, 158630641, 177975881, 199691526, 870287321
Offset: 1

Views

Author

Shanzhen Gao, Sep 19 2013

Keywords

Comments

For n = 1, 2, 3, 4, 6, 7, 10, 12, 16, 18, 23, a(n) is in A049685.

Examples

			a(1)=1 and a(2)=6 since (1, 1, 1, 1, 1, 1, 1) and (6, 1, 1, 1, 1, 1, 1) satisfying a^2+b^2+c^2+d^2+e^2+f^2+g^2=7abcdefg.

Crossrefs

Cf. A049685, A000032.
Cf. A227210, A227211, A227212.

A227211 Generalized Markoff numbers: largest of 6-tuple of positive numbers a, b, c, d, e, f, g satisfying the Markoff(7) equation a^2+b^2+c^2+d^2+e^2+f^2+g^2 = 3abcdefg.

Original entry on oeis.org

3, 7, 17, 18, 47, 62, 99, 123, 151, 305, 322, 377, 551, 577, 843, 1299, 1342, 2207, 2537, 3363, 3905, 4897, 5047, 5473, 5778, 7698, 7899, 10097, 11927, 15127, 17342, 19601, 20351, 23102, 27217, 31107, 39603, 43522, 46663, 93329, 98209
Offset: 1

Views

Author

Shanzhen Gao, Sep 19 2013

Keywords

Examples

			a(1)=3 and a(2)=7 since (3, 2, 1, 1, 1, 1, 1) and (7, 3, 1, 1, 1, 1, 1) satisfying a^2+b^2+c^2+d^2+e^2+f^2+g^2=3abcdefg with a>=b>=c>=d>=e>=f>=g. a(n) is the first components among all solutions in increasing order.

Crossrefs

Cf. A227210, A227212, A227214.

A227210 Generalized Markoff numbers: largest of 7-tuple of positive numbers a, b, c, d, e, f, g satisfying the Markoff(7) equation a^2+b^2+c^2+d^2+e^2+f^2+g^2 = 2abcdefg.

Original entry on oeis.org

2, 6, 15, 22, 47, 82, 118, 239, 262, 306, 370, 527, 929, 1126, 1142, 2255, 2913, 3122, 3606, 3809, 4262, 7078, 7215, 7314, 11522, 12626, 14159, 15906, 22934, 26977, 28678
Offset: 1

Views

Author

Shanzhen Gao, Sep 19 2013

Keywords

Examples

			a(1)=2 and a(2)=6 since (2, 2, 2, 1, 1, 1, 1) and (6, 2, 2, 1, 1, 1, 1) satisfying a^2+b^2+c^2+d^2+e^2+f^2+g^2=2abcdefg with a>=b>=c>=d>=e>=f>=g. a(n) is the first components among all solutions in increasing order.

Crossrefs

Cf. A227211, A227212, A227214.

A227204 Largest number in a 6-tuple (a,b,c,d,e,f) of positive integers satisfying the Markoff(6) equation a^2 + b^2 + c^2 + d^2 + e^2 + f^2 = 3*a*b*c*d*e*f.

Original entry on oeis.org

2, 4, 10, 11, 23, 26, 64, 68, 119, 131, 134, 178, 274, 373, 466, 551, 779, 781, 1220, 1418, 1561, 2110, 2174, 3194, 3265, 3566, 4223, 4552, 5303, 8362, 8644, 9244, 12671, 14279, 16897, 17291, 18491, 18601, 18902, 21892, 23344, 26531, 36311, 38906, 57314, 60752, 69566, 71614, 73852
Offset: 1

Views

Author

Shanzhen Gao, Sep 18 2013

Keywords

Examples

			2 is in the sequence since (2, 2, 1, 1, 1, 1) is a solution to a^2 + b^2 + c^2 + d^2 + e^2 + f^2 = 3 *a*b*c*d*e*f. 4, 10, and 11 are in the sequence since (4, 2, 1, 1, 1, 1), (10, 4, 1, 1, 1, 1), (11, 2, 2, 1, 1, 1) are solutions with a>=b>=c>=d>=e>=f>=g.  This sequence lists the first components among all solutions in increasing order.

Links

Crossrefs

Cf. A002559.

A227208 Largest number in an integer 7-tuple (a, b, c, d, e, f, g) satisfying the Markoff(7) equation a^2+b^2+c^2+d^2+e^2+f^2+g^2 = a*b*c*d*e*f*g.

Original entry on oeis.org

3, 5, 10, 18, 23, 37, 39, 58, 67, 119, 138, 178, 181, 250, 274, 307, 338, 359, 515, 551, 738, 778, 933, 1157, 1418, 1425, 1479, 1559, 1738, 1762, 1922, 1970, 2410, 2417, 3265
Offset: 1

Views

Author

Shanzhen Gao, Sep 19 2013

Keywords

Examples

			3 and 5 are in the sequence since (3, 2, 2, 2, 1, 1, 1) and (5, 2, 2, 2, 1, 1, 1) satisfy a^2+b^2+c^2+d^2+e^2+f^2+g^2 = a*b*c*d*e*f*g with a >= b >= c >= d >= e >= f >= g.

Crossrefs

Cf. A002559.

A229241 Generalized Markoff numbers: largest number a in a 5-tuple a >= b >= c >= d >= e satisfying the Markoff(5) equation a^2 + b^2 + c^2 + d^2 + e^2 = 4*a*b*c*d*e.

Original entry on oeis.org

2, 7, 26, 55, 97, 362, 433, 727, 1351, 1538, 3079, 3409, 5042, 10087, 18817, 20330, 26839, 43009, 70226, 75601, 95258, 140455, 172369, 190519, 211303, 262087, 338353, 529255, 568513, 978122, 1048951, 1202714, 1354753, 1663585, 1956247, 2368519, 3650401, 3913730, 4741658, 5904386, 7861777, 9483319
Offset: 1

Views

Author

Shanzhen Gao, Sep 17 2013

Keywords

Examples

			a(1)=2 since (2, 1, 1, 1, 1) is a solution of a^2 + b^2 + c^2 + d^2 + e^2 = 4*a*b*c*d*e. a(2)=7 since (7, 2, 1, 1, 1) is a solution. a(3)=26 since (26,7,1,1,1) is a solution.

A227206 Largest number in an integer 6-tuple (a, b, c, d, e, f) satisfying the Markoff(6) equation a^2+b^2+c^2+d^2+e^2+f^2 = 6*a*b*c*d*e*f.

Original entry on oeis.org

1, 5, 29, 169, 869, 985, 5741, 26041, 29405, 33461, 151201, 195025, 756029, 780361, 998789, 1136689, 5116301, 6625109, 23384789, 26308105, 29816641, 33929309, 38613965, 135777769, 225058681, 657744361, 678888869, 700763309, 788361985, 864683429, 890206969, 1012771061, 1152597605
Offset: 1

Views

Author

Shanzhen Gao, Sep 18 2013

Keywords

Comments

2*a(n)^2-1 is a square for a(1), a(2), a(3), a(4), a(6), a(7), a(10), a(12), a(16), a(18), a(23), a(25),...

Examples

			1 is in the sequence since (1, 1, 1, 1, 1, 1) is a solution to a^2+b^2+c^2+d^2+e^2+f^2 = 6*a*b*c*d*e*f. 5, 29, and 169 are in the sequence since (5, 1, 1, 1, 1, 1), (29, 5, 1, 1, 1, 1), (169, 29, 1, 1, 1, 1) are solutions.

Crossrefs

Cf. A001653, A229242 (Markoff(5)).

A229242 Generalized Markoff numbers: union of numbers a, b, c, d, e satisfying the Markoff(5) equation a^2 + b^2 + c^2 + d^2 + e^2 = 5*a*b*c*d*e.

Original entry on oeis.org

1, 4, 19, 91, 379, 436, 2089, 7561, 8644, 10009, 36001, 47956, 144019, 150841, 198379, 229771, 821089, 1100899, 3009259, 3419716, 3933001, 4554019, 5274724, 14328091, 25272721, 54726841, 57312379, 60034339, 68221876, 74727379, 77994811, 90262009, 104544004
Offset: 1

Views

Author

Shanzhen Gao, Sep 17 2013

Keywords

Examples

			a(1)=1 since (1, 1, 1, 1, 1) is a solution to a^2 + b^2 + c^2 + d^2 + e^2 = 5*a*b*c*d*e. a(2)=4 since (4, 1, 1, 1, 1) is a solution. a(3)=19 since (19,4,1,1,1) is a solution. a(n) is the n-th leading term (that is a's) among all the solutions with a >= b >= c >= d >= e.

A229240 Sorted list of largest values in the quintuple (a,b,c,d,e) satisfying a^2 + b^2 + c^2 + d^2 + e^2 = a*b*c*d*e.

Original entry on oeis.org

4, 5, 9, 12, 23, 31, 33, 35, 44, 57, 60, 81, 107, 123, 157, 179, 204, 212, 273, 293, 311, 369, 391, 411, 417, 459, 555, 620, 657, 679, 1076, 1115, 1187, 1259, 1275, 1308, 1377, 1453, 1713, 1813, 1979, 2508, 2604, 2673, 2764, 2817, 2885, 3419, 3475, 3804, 3849, 4055, 4139, 4397, 4683, 4969, 6267, 6393, 6439, 7375
Offset: 1

Views

Author

Shanzhen Gao, Sep 17 2013

Keywords

Examples

			a(1)=4 since (4, 3, 3, 1, 1) is a solution of a^2 + b^2 + c^2 + d^2 + e^2 = a*b*c*d*e.
a(2)=5 since (5, 3, 3, 1, 1) is a solution.
a(3)=9 since (9, 4, 3, 1, 1) is a solution.

Extensions

Definition corrected by Ralf Stephan, Sep 18 2013

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