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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.

%I A210503 #34 Jan 09 2025 18:29:20
%S A210503 15,35,143,323,899,1763,3599,4641,5183,10403,11663,13585,19043,22499,
%T A210503 32399,35581,36863,39203,51983,57599,72899,79523,97343,121103,176399,
%U A210503 186623,213443,272483,324899,359999,381923,412163,435599,446641,622081,656099,675683
%N 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.
%C A210503 A037074 is a subsequence of this sequence.
%C A210503 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 A210503 Also in the sequence are the following numbers with four distinct prime factors:
%C A210503         4641 = 3*7*13*17       [form p(p+4)*q(q+4)],
%C A210503        13585 = 5*11*13*19      [form p(p+6)*q(q+6)],
%C A210503        35581 = 7*13*17*23      [form p(p+6)*q(q+6)],
%C A210503       446641 = 13*17*43*47     [form p(p+4)*q(q+4)],
%C A210503       622081 = 17*23*37*43     [form p(p+6)*q(q+6)],
%C A210503       700321 = 19*29*31*41     [form p(p+10)*q(q+10)],
%C A210503 From _Ray Chandler_, Jan 25 2017: (Start)
%C A210503     24887581 = 47*53*97*103    [form p(p+6)*q(q+6)],
%C A210503     43518577 = 59*67*101*109   [form p(p+8)*q(q+8)],
%C A210503    115539901 = 83*97*113*127   [form p(p+14)*q(q+14)],
%C A210503    158682817 = 89*101*127*139  [form p(p+12)*q(q+12)],
%C A210503    305162941 = 103*113*157*167 [form p(p+10)*q(q+10)],
%C A210503   1093514641 = 103*107*313*317 [form p(p+4)*q(q+4)],
%C A210503   1415940061 = 167*193*197*223 [form p(p+26)*q(q+26)].
%C A210503 And one term with six distinct prime factors:
%C A210503    650344079 = 7*11*37*53*59*73. (End)
%H A210503 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 A210503 m=57599, m'=480, sqrt(57599^2 + 480^2) = 57601.
%p A210503 with(numtheory);
%p A210503 A210503:= proc(q)
%p A210503 local a,n,p;
%p A210503 for n from 1 to q do
%p A210503   a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
%p A210503   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 A210503 od; end:
%p A210503 A210503(100000);
%o A210503 (Python)
%o A210503 from math import sqrt
%o A210503 from sympy import factorint
%o A210503 from gmpy2 import mpz, is_square, gcd
%o A210503 A210503 = []
%o A210503 for n in range(2, 10**5):
%o A210503     nd = sum([mpz(n*e/p) for p, e in factorint(n).items()])
%o A210503     if is_square(nd**2+n**2) and gcd(gcd(n, nd), mpz(sqrt(nd**2+n**2))) == 1:
%o A210503         A210503.append(n) # _Chai Wah Wu_, Aug 21 2014
%Y A210503 Cf. A003415, A009003, A009004, A037074.
%K A210503 nonn
%O A210503 1,1
%A A210503 _Paolo P. Lava_, Jan 25 2013