A396672 - OEIS

A396672

Averages of two consecutive odd squarefree semiprimes p, q (A046388) that share a common factor > 1 with at least one of p and q.

%I #11 Jun 07 2026 23:23:59

%S 18,27,45,126,150,180,360,540,576,594,810,846,936,1296,1323,2466,2646,

%T 2850,3000,3060,3186,3300,3690,3834,3906,3915,4050,4356,4446,4464,

%U 4653,4797,4824,5004,5232,5796,5850,6150,6264,6336,6390,6426,6450,6600,6714,6723,7020

%N Averages of two consecutive odd squarefree semiprimes p, q (A046388) that share a common factor > 1 with at least one of p and q.

%C The least terms not divisible by 3 are 127250, 494750, 578830, ... .

%H Karl-Heinz Hofmann, <a href="/A396672/b396672.txt">Table of n, a(n) for n = 1..10000</a>

%e j A046388(j)

%e (A046388(j)+A046388(j+1))/2

%e Factorizations

%e Terms

%e 1 15 3 * 5

%e 18 2 * 3^2 a(1) = 18

%e 2 21 3 * 7

%e 27 3^3 a(2) = 27

%e 3 33 3 * 11

%e 34 2 * 17

%e 4 35 5 * 7

%e 37 37

%e 5 39 3 * 13

%e 45 3^2 * 5 a(3) = 45

%e 6 51 3 * 17

%e 53 53

%e 7 55 5 * 11

%e ...

%e 127235 5 * 25447

%e 127250 2 * 5^3 * 509 a(..) = 127250

%e 127265 5 * 25453

%o (PARI) a396672(upto=7500) = my(p1=15,p2); forstep(k=21, upto, 2, my(f=factor(k), m); if(omega(f)==2 && bigomega(f)==2, p2=k; m=p1+p2; if(gcd(p1,m)!=1 || gcd(p2,m)!=1, print1(m/2, ", ")); p1=p2))

%o (Python)

%o from sympy import factorint

%o def A046388_isok_and_factorization(n):

%o return [n % 2 == 1 and (len(f := factorint(n,multiple=True)) == 2 and f[0] != f[1]), f]

%o A046388, A396672, k = [15], [], 19

%o while len(A396672) < (aupto := 47):

%o k += 2

%o if (r:=A046388_isok_and_factorization(k))[0]:

%o A046388.append(k)

%o if (f := factorint(A046388[-2],multiple=True))[0] in r[1] or f[1] in r[1]:

%o A396672.append((A046388[-2] + A046388[-1]) // 2)

%o print(A396672) # _Karl-Heinz Hofmann_, Jun 02 2026

%Y Cf. A046388.

%K nonn

%O 1,1

%A _Hugo Pfoertner_, Jun 02 2026