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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.
7
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
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Jan 25 2013
STATUS
approved