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A348077 Starts of runs of 3 consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039). 3
603, 1250, 1323, 2523, 4203, 4923, 4948, 7442, 10467, 12591, 18027, 20402, 21123, 23823, 31507, 31850, 36162, 40327, 54475, 54511, 55323, 58923, 63747, 64386, 71523, 73204, 79011, 83151, 85291, 88047, 97675, 103923, 104211, 118323, 120787, 122571, 124891, 126927 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

EXAMPLE

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.

MATHEMATICA

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

PROG

(Python)

from sympy import factorint

def aupto(limit):

    alst, condvec = [], [False, False, False]

    for kp2 in range(4, limit+3):

        evenodd = [0, 0]

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

            evenodd[e%2] += 1

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

        if all(condvec):

            alst.append(kp2-2)

    return alst

print(aupto(126927)) # Michael S. Branicky, Sep 27 2021

CROSSREFS

Subsequence of A187039 and A348076.

Sequence in context: A218521 A145331 A252957 * A066785 A178032 A107256

Adjacent sequences:  A348074 A348075 A348076 * A348078 A348079 A348080

KEYWORD

nonn

AUTHOR

Amiram Eldar, Sep 27 2021

STATUS

approved

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Last modified December 6 13:45 EST 2021. Contains 349563 sequences. (Running on oeis4.)