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

1,1

Comments

First differs from A049103 and A074172 at n=7.

Links

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Subsequence of A187039.

A074172 is a subsequence.

Cf. A049103.

Sequence in context: A156812 A171665 A348098 * A348345 A049103 A074172

Adjacent sequences: A348073 A348074 A348075 * A348077 A348078 A348079

Keyword

nonn

Author

Amiram Eldar, Sep 27 2021

Status

approved