A210503 Numbers k that form a primitive Pythagorean triple with k' and sqrt(k^2 + k'^2), where k' is the arithmetic derivative of k.

15, 35, 143, 323, 899, 1763, 3599, 4641, 5183, 10403, 11663, 13585, 19043, 22499, 32399, 35581, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 446641, 622081, 656099, 675683

Offset

1,1

Comments

A037074 is a subsequence of this sequence.

If k is the product of a pair of twin primes we have k=p(p+2), k'=2(p+1) and sqrt(k^2+k'^2)=(p+1)^2+1=p(p+2)+2=k+2. These numbers are relatively prime and therefore they form a primitive Pythagorean triple.

Also in the sequence are the following numbers with four distinct prime factors:

4641 = 3*7*13*17 [form p(p+4)*q(q+4)],

13585 = 5*11*13*19 [form p(p+6)*q(q+6)],

35581 = 7*13*17*23 [form p(p+6)*q(q+6)],

446641 = 13*17*43*47 [form p(p+4)*q(q+4)],

622081 = 17*23*37*43 [form p(p+6)*q(q+6)],

700321 = 19*29*31*41 [form p(p+10)*q(q+10)],

From Ray Chandler, Jan 25 2017: (Start)

24887581 = 47*53*97*103 [form p(p+6)*q(q+6)],

43518577 = 59*67*101*109 [form p(p+8)*q(q+8)],

115539901 = 83*97*113*127 [form p(p+14)*q(q+14)],

158682817 = 89*101*127*139 [form p(p+12)*q(q+12)],

305162941 = 103*113*157*167 [form p(p+10)*q(q+10)],

1093514641 = 103*107*313*317 [form p(p+4)*q(q+4)],

1415940061 = 167*193*197*223 [form p(p+26)*q(q+26)].

And one term with six distinct prime factors:

650344079 = 7*11*37*53*59*73. (End)

Links

Ray Chandler, Table of n, a(n) for n = 1..500 (terms 1..100 from Paolo P. Lava)

Example

m=57599, m'=480, sqrt(57599^2 + 480^2) = 57601.

Maple

with(numtheory);
A210503:= proc(q)
local a, n, p;
for n from 1 to q do
  a:=n*add(op(2, p)/op(1, p), p=ifactors(n)[2]);
  if trunc(sqrt(n^2+a^2))=sqrt(n^2+a^2) and gcd(n, gcd(a, n^2+a^2))=1 then print(n); fi;
od; end:
A210503(100000);

Prog

(Python)
from sympy import factorint
from gmpy2 import mpz, is_square, gcd
A210503 = []
for n in range(2, 10**5):
....nd = sum([mpz(n*e/p) for p, e in factorint(n).items()])
....if is_square(nd**2+n**2) and gcd(gcd(n, nd), mpz(sqrt(nd**2+n**2))) == 1:
........A210503.append(n) # Chai Wah Wu, Aug 21 2014

Crossrefs

Cf. A003415, A009003, A009004, A037074.

Sequence in context: A254031 A074480 A194580 * A037074 A107423 A027442

Adjacent sequences: A210500 A210501 A210502 * A210504 A210505 A210506

Keyword

nonn

Author

Paolo P. Lava, Jan 25 2013

Status

approved