A348077 Starts of runs of 3 consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039).

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

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