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A348076 Number k such that k and k+1 both have an equal number of even and odd exponents in their prime factorization (A187039).

%I A348076 #13 Sep 28 2021 08:33:46
%S A348076 44,75,98,116,147,171,175,207,244,332,368,387,404,507,548,603,604,656,
%T A348076 724,800,832,844,847,891,908,931,963,1052,1075,1083,1124,1250,1251,
%U A348076 1323,1324,1412,1467,1556,1587,1675,1772,1791,2096,2224,2312,2348,2367,2511,2523
%N A348076 Number k such that k and k+1 both have an equal number of even and odd exponents in their prime factorization (A187039).
%C A348076 First differs from A049103 and A074172 at n=7.
%H A348076 Amiram Eldar, <a href="/A348076/b348076.txt">Table of n, a(n) for n = 1..10000</a>
%e A348076 44 is a term since 44 = 2^2 * 11 and 44 + 1 = 45 = 3^2 * 5 both have one even and one odd exponent in their prime factorization.
%t A348076 q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), _?OddQ] == Count[e, _?EvenQ]; Select[Range[2500], q[#] && q[# + 1] &]
%o A348076 (Python)
%o A348076 from sympy import factorint
%o A348076 def aupto(limit):
%o A348076     alst, cond = [], False
%o A348076     for nxtk in range(3, limit+2):
%o A348076         evenodd = [0, 0]
%o A348076         for e in factorint(nxtk).values():
%o A348076             evenodd[e%2] += 1
%o A348076         nxtcond = (evenodd[0] == evenodd[1])
%o A348076         if cond and nxtcond:
%o A348076             alst.append(nxtk-1)
%o A348076         cond = nxtcond
%o A348076     return alst
%o A348076 print(aupto(2523)) # _Michael S. Branicky_, Sep 27 2021
%Y A348076 Subsequence of A187039.
%Y A348076 A074172 is a subsequence.
%Y A348076 Cf. A049103.
%K A348076 nonn
%O A348076 1,1
%A A348076 _Amiram Eldar_, Sep 27 2021