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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). 5

%I #13 Sep 28 2021 08:33:46

%S 44,75,98,116,147,171,175,207,244,332,368,387,404,507,548,603,604,656,

%T 724,800,832,844,847,891,908,931,963,1052,1075,1083,1124,1250,1251,

%U 1323,1324,1412,1467,1556,1587,1675,1772,1791,2096,2224,2312,2348,2367,2511,2523

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

%C First differs from A049103 and A074172 at n=7.

%H Amiram Eldar, <a href="/A348076/b348076.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), _?OddQ] == Count[e, _?EvenQ]; Select[Range[2500], q[#] && q[# + 1] &]

%o (Python)

%o from sympy import factorint

%o def aupto(limit):

%o alst, cond = [], False

%o for nxtk in range(3, limit+2):

%o evenodd = [0, 0]

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

%o evenodd[e%2] += 1

%o nxtcond = (evenodd[0] == evenodd[1])

%o if cond and nxtcond:

%o alst.append(nxtk-1)

%o cond = nxtcond

%o return alst

%o print(aupto(2523)) # _Michael S. Branicky_, Sep 27 2021

%Y Subsequence of A187039.

%Y A074172 is a subsequence.

%Y Cf. A049103.

%K nonn

%O 1,1

%A _Amiram Eldar_, Sep 27 2021

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Last modified April 15 20:47 EDT 2024. Contains 371696 sequences. (Running on oeis4.)