A348076 Number k such that k and k+1 both have an equal number of even and odd exponents in their prime factorization (A187039).

44, 75, 98, 116, 147, 171, 175, 207, 244, 332, 368, 387, 404, 507, 548, 603, 604, 656, 724, 800, 832, 844, 847, 891, 908, 931, 963, 1052, 1075, 1083, 1124, 1250, 1251, 1323, 1324, 1412, 1467, 1556, 1587, 1675, 1772, 1791, 2096, 2224, 2312, 2348, 2367, 2511, 2523

Offset: 1

Author

Amiram Eldar, Sep 27 2021

Keywords

• nonn

Comments

First differs from A049103 and A074172 at n=7.

Examples

			44 is a term since 44 = 2^2 * 11 and 44 + 1 = 45 = 3^2 * 5 both have one even and one odd exponent in their prime factorization.
		

Links

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

Crossrefs

Subsequence of A187039.

A074172 is a subsequence.

Cf. A049103.

Programs

Mathematica

q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), ?OddQ] == Count[e, ?EvenQ]; Select[Range[2500], q[#] && q[# + 1] &]

Python

from sympy import factorint
def aupto(limit):
    alst, cond = [], False
    for nxtk in range(3, limit+2):
        evenodd = [0, 0]
        for e in factorint(nxtk).values():
            evenodd[e%2] += 1
        nxtcond = (evenodd[0] == evenodd[1])
        if cond and nxtcond:
            alst.append(nxtk-1)
        cond = nxtcond
    return alst
print(aupto(2523)) # Michael S. Branicky, Sep 27 2021