A348078 Starts of runs of 4 consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039).

906596, 1141550, 1243275, 12133673, 13852924, 19293209, 20738672, 22997761, 23542001, 26587348, 30731822, 31237450, 39987773, 41419024, 43627148, 54040975, 54652148, 56487148, 70289225, 75855625, 77449300, 79677772, 80665072, 82126448, 91420721, 93883850, 95162849

Offset: 1

Author

Amiram Eldar, Sep 27 2021

Keywords

• nonn

Examples

			906596 is a term since 906596 = 2^2 * 226649, 906596 + 1 = 906597 = 3^2 * 100733, 906596 + 2 = 906598 = 2 * 7^2 * 11 * 29^2 and 906596 + 3 = 906599 = 71 * 113^2 all have an equal number of even and odd exponents in their prime factorization.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Subsequence of A187039, A348076 and A348077.

Programs

Mathematica

q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), ?OddQ] == Count[e, ?EvenQ]; v = q /@ Range[4]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 3]], {k, 5, 2*10^7}]; seq

Python

from sympy import factorint
def cond(n):
    evenodd = [0, 0]
    for e in factorint(n).values():
        evenodd[e%2] += 1
    return evenodd[0] == evenodd[1]
def afind(limit, startk=5):
    condvec = [cond(startk+i) for i in range(4)]
    for kp3 in range(startk+3, limit+4):
        condvec = condvec[1:] + [cond(kp3)]
        if all(condvec):
            print(kp3-3, end=", ")
afind(125*10**4) # Michael S. Branicky, Sep 27 2021