A348077 - OEIS

%I #11 Sep 28 2021 08:34:01

%S 603,1250,1323,2523,4203,4923,4948,7442,10467,12591,18027,20402,21123,

%T 23823,31507,31850,36162,40327,54475,54511,55323,58923,63747,64386,

%U 71523,73204,79011,83151,85291,88047,97675,103923,104211,118323,120787,122571,124891,126927

%N Starts of runs of 3 consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039).

%H Amiram Eldar, <a href="/A348077/b348077.txt">Table of n, a(n) for n = 1..10000</a>

%e 603 is a term since 603 = 3^2 * 67, 603 + 1 = 604 = 2^2 * 151 and 603 + 2 = 605 = 5 * 11^2 all have one even and one odd exponent in their prime factorization.

%t q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), _?OddQ] == Count[e, _?EvenQ]; v = q /@ Range[3]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 2]], {k, 4, 130000}]; seq

%o (Python)

%o from sympy import factorint

%o def aupto(limit):

%o     alst, condvec = [], [False, False, False]

%o     for kp2 in range(4, limit+3):

%o         evenodd = [0, 0]

%o         for e in factorint(kp2).values():

%o             evenodd[e%2] += 1

%o         condvec = condvec[1:] + [evenodd[0] == evenodd[1]]

%o         if all(condvec):

%o             alst.append(kp2-2)

%o     return alst

%o print(aupto(126927)) # _Michael S. Branicky_, Sep 27 2021

%Y Subsequence of A187039 and A348076.

%K nonn

%O 1,1

%A _Amiram Eldar_, Sep 27 2021