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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 subsequence 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.

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

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, 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

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Last modified May 25 23:41 EDT 2017. Contains 287073 sequences.