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

1,1

Links

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

Example

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.

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

Prog

(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

Crossrefs

Subsequence of A187039, A348076 and A348077.

Sequence in context: A251520 A250542 A251799 * A348101 A179735 A015333

Adjacent sequences: A348075 A348076 A348077 * A348079 A348080 A348081

Keyword

nonn

Author

Amiram Eldar, Sep 27 2021

Status

approved