A210503 - OEIS

%I #31 Feb 18 2021 00:45:22

%S 15,35,143,323,899,1763,3599,4641,5183,10403,11663,13585,19043,22499,

%T 32399,35581,36863,39203,51983,57599,72899,79523,97343,121103,176399,

%U 186623,213443,272483,324899,359999,381923,412163,435599,446641,622081,656099,675683

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

%C A037074 is a subsequence of this sequence.

%C 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.

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

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

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

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

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

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

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

%C From _Ray Chandler_, Jan 25 2017: (Start)

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

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

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

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

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

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

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

%C And one term with six distinct prime factors:

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

%H Ray Chandler, <a href="/A210503/b210503.txt">Table of n, a(n) for n = 1..500</a> (terms 1..100 from Paolo P. Lava)

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

%p with(numtheory);

%p A210503:= proc(q)

%p local a,n,p;

%p for n from 1 to q do

%p   a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);

%p   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;

%p od; end:

%p A210503(100000);

%o (Python)

%o from sympy import factorint

%o from gmpy2 import mpz, is_square, gcd

%o A210503 = []

%o for n in range(2,10**5):

%o ....nd = sum([mpz(n*e/p) for p,e in factorint(n).items()])

%o ....if is_square(nd**2+n**2) and gcd(gcd(n,nd),mpz(sqrt(nd**2+n**2))) == 1:

%o ........A210503.append(n) # _Chai Wah Wu_, Aug 21 2014

%Y Cf. A003415, A009003, A009004, A037074.

%K nonn

%O 1,1

%A _Paolo P. Lava_, Jan 25 2013