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A337140 Numbers m = a + b with a and b positive integers whose product a*b = k^2 is a square.

%I A337140 #39 Aug 22 2024 02:43:34
%S A337140 2,4,5,6,8,10,12,13,14,15,16,17,18,20,22,24,25,26,28,29,30,32,34,35,
%T A337140 36,37,38,39,40,41,42,44,45,46,48,50,51,52,53,54,55,56,58,60,61,62,64,
%U A337140 65,66,68,70,72,73,74,75,76,78,80,82,84,85,86,87,88,89,90,91
%N A337140 Numbers m = a + b with a and b positive integers whose product a*b = k^2 is a square.
%C A337140 Related to Heron triangles with a partition point on one of the sides. Calculations become quite different when the partition a + b = m gives the perfect square k^2 = a*b.
%C A337140 These numbers coincide with the numbers > 1 not in A004614.
%C A337140 Let m = 2^t * p_1^a_1 * p_2^a_2 * ... * p_r^a_r * q_1^b_1 * q_2^b_2 * ... * q_s^b_s with t >= 0, a_i >= 0 for i=1..r, where p_i == 1 (mod 4) for i=1..r and q_j == -1 (mod 4) for j=1..s.
%C A337140 Even numbers (A005843) belong to this sequence: m = 2*k and p = k^2.
%C A337140 Numbers divisible by a prime q congruent to 1 (mod 4) (cf. A004613) belong to this sequence: m = q * m_1 = (u^2 + v^2) * m_1 and p = (u*v*q)^2.
%C A337140 The other numbers are divisible only by primes congruent to 3 (mod 4) (cf. A004614).
%C A337140 If a term m is not in the union of A005843 and A004613, then m = q_1^b_1 * q_2^b_2 * ... * q_s^b_s is a term of A018825 (numbers not the sum of two nonzero squares) = q_i * m_1 = q_i *(u^2 + v) and p = q_i^2 * u^2 * v for all u^2 < m_1 and v nonsquare. And so m is not a term: A contradiction.
%e A337140 Even numbers m = 2*k give a = b = k. For example, 94 = 47+47 and k^2 = 47^2.
%e A337140 Numbers which are divisible by a prime q congruent to 1 (mod 4) give m = q*m' = (u^2 + v^2)*m' and p = (u*v*m')^2. For example, 87 = 3*29 = 3*(25 + 4) = (5*4*3)^2 = 60^2.
%t A337140 Select[Range[100], Length @ Select[Times @@@ IntegerPartitions[#, {2}], IntegerQ @ Sqrt[#1] &] > 0 &] (* _Amiram Eldar_, Aug 26 2020 *)
%o A337140 (PARI) upto(n) = { my(res = List(vector(n\2, i, 2*i))); forstep(i = 1, n, 2, c = core(i); for(k = 1, sqrtint((n-i)\c), listput(res, i + c*k^2); ) ); listsort(res, 1); res } \\ _David A. Corneth_, Aug 26 2020
%o A337140 (PARI) is(n) = for(i = 1, n\2 + 1, if(issquare(i * (n-i)), return(n>1))); 0 \\ _David A. Corneth_, Aug 26 2020
%o A337140 (Python)
%o A337140 from itertools import count, islice
%o A337140 from sympy import primefactors
%o A337140 def A337140_gen(startvalue=2): # generator of terms >= startvalue
%o A337140     return filter(lambda n: n&1^1 or not all(p&2 for p in primefactors(n>>(~n & n-1).bit_length())), count(max(startvalue,2)))
%o A337140 A337140_list = list(islice(A337140_gen(),30)) # _Chai Wah Wu_, Aug 21 2024
%Y A337140 Cf. A004614.
%Y A337140 Cf. A004613, A005843, A018825.
%K A337140 nonn,easy
%O A337140 1,1
%A A337140 _Hein van Winkel_, Aug 18 2020