A348079 Starts of runs of 5 consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039).

792007675, 2513546971, 2820448771, 3201296272, 4742326672, 4894282924, 5462510272, 5664816448, 6947006272, 7814337424, 8784450448, 9085360624, 10147712524, 10246365547, 11537724975, 11861786572, 11907710548, 12456672496, 13338112048, 13510075471, 13931933948

Offset

1,1

Links

Table of n, a(n) for n=1..21.

Example

792007675 is a term since 792007675 = 2^2 * 31680307, 792007675 + 1 = 792007676 = 2^2 * 198001919, 792007675 + 2 = 792007677 = 3^2 * 88000853, 792007675 + 3 = 792007678 = 2 * 7^2 * 11^2 * 66791 and 792007675 + 4 = 792007679 = 17^2 * 2740511 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[5]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 4]], {k, 6, 3*10^9}]; 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=6):
    condvec = [cond(startk+i) for i in range(5)]
    for kp4 in range(startk+4, limit+5):
        condvec = condvec[1:] + [cond(kp4)]
        if all(condvec):
            print(kp4-4, end=", ")
afind(10**9) # Michael S. Branicky, Sep 27 2021

Crossrefs

Subsequence of A187039, A348076, A348077 and A348078.

Sequence in context: A344730 A225389 A166121 * A046186 A166227 A104829

Adjacent sequences: A348076 A348077 A348078 * A348080 A348081 A348082

Keyword

nonn

Author

Amiram Eldar, Sep 27 2021

Status

approved