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A210503 Numbers n that form a primitive Pythagorean triple with n’ and sqrt(n^2 + n’^2), where n’ is the arithmetic derivative of n. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A037074 is a subset of this sequence.

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

In the sequence we have also numbers like:

4641= 3*7*13*17.     [of the form p1(p1+4)*p2(p2+4)]

13585= 5*11*13*19.   [of the form p1(p1+6)*p2(p2+6)]

35581= 7*13*17*23.   [of the form p1(p1+6)*p2(p2+6)]

446641= 13*17*43*47. [of the form p1(p1+4)*p2(p2+4)]

622081= 17*23*37*43. [of the form p1(p1+6)*p2(p2+6)]

700321= 19*29*31*41. [of the form p1(p1+10)*p2(p2+10)]

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..100

EXAMPLE

n=57599, n’=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.

Sequence in context: A142591 A074480 A194580 * A037074 A107423 A027442

Adjacent sequences:  A210500 A210501 A210502 * A210504 A210505 A210506

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Jan 25 2013

STATUS

approved

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Last modified December 19 05:27 EST 2014. Contains 252177 sequences.