A277534 Least hypotenuse, c, of a Primitive Pythagorean Triangle (PPT) such that the difference between it, c, and its greater leg, b, is n; or 0 if no such PPT exists.

5, 17, 0, 0, 65, 0, 0, 29, 65, 185, 0, 0, 169, 0, 0, 0, 221, 333, 0, 0, 273, 0, 0, 0, 157, 481, 0, 0, 1189, 0, 0, 641, 1353, 629, 0, 0, 1517, 0, 0, 425, 1681, 777, 0, 0, 1845, 0, 0, 0, 205, 925, 0, 0, 2173, 0, 0, 0, 2337, 1073, 0, 0, 2501, 0, 0, 0, 2665, 1221, 0, 0, 2829, 0, 0, 1405, 2993, 1369, 0

Offset: 1

Author

Ron Knott and Robert G. Wilson v, Jun 05 2014

Keywords

• nonn
• easy

Comments

n = 1, 2, 5, 8, 9, 10, 13, 17, 18, 21, 25, ..., satisfies the first criterion;

a(n) = 0 for n = 3, 4, 6, 7, 11, 12, 14, 15, 16, 19, 20, 22, 23, 24, ..., ;

a(n) = 0 for 5832 of the first 10000 terms;

a(8n) = 0 for 832 of the first 10000 terms;

a(8n) = 0 for n: 2, 3, 6, 7, 8, 10, 11, 12, 14, 15, 18, 19, 22, 23, 24, ..., ;

a(8n+1) > 0;

a(8n+2) > 0; a linear 2nd-order recurrence: a(n) = 2*a(n-1) - a(n-2) with a(1) = 185 & a(2) = 333;

a(8n+3) = 0;

a(8n+4) = 0;

a(8n+5) > 0;

a(8n+6) = 0;

a(8n+7) = 0;

Prime terms: 5, 17, 29, 157, 641, 3821, 4201, 17749, 21601, 31981, 38273, 44789, 61129, 66173, 72161, 100673, 108541, 114553, 121421, 142973, 165541, 173777, 182141, 204733, 213881, 225889, 235493, 281837, ..., .

Examples

			a(1) is 5 since the PPT (3,4,5) satisfies the first stated criterion; a(2) is 17 since the PPT (8,15,17) satisfies the first stated criterion; a(3) = 0 since there exists no PPT that satisfies the stated criteria; etc.
		

Links

• Ron Knott and Robert G. Wilson v, Table of n, a(n) for n = 1..10000
• Ron Knott, Pythagorean Triples and Online Calculators

Crossrefs

Cf. A008846, A020882, A242219.

Programs

Mathematica

f[n_] := FindInstance[ a^2 + b^2 == c^2 && Mod[c, 4] == 1 && 0 < a < b < c && c - b == n, {a, b, c}, Integers][[1, 3, 2, 1, 1, 3]] + 1 /. 1 + {}[[1, 3, 2, 1, 1, 3]] -> 0; f[1] = 5; Array[f, 75]