A355822 Numbers k such that A003961(k) and A276086(k) share a prime factor, where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 92, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115

Offset: 1

Author

Antti Karttunen, Jul 18 2022

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to primorial base

Crossrefs

Positions of terms > 1 in A355442 and in A355001.

Cf. A003961, A276086, A355002 (subsequence), A355820 (positions of zeros), A355821 (complement), A355835.

Cf. A005843 (even numbers, apart from 0, is a subsequence).

Cf. also A324584.

Programs

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355820(n) = (1==gcd(A003961(n), A276086(n)));
isA355822(n) = !A355820(n);

Python

from math import prod, gcd
from itertools import count, islice
from sympy import nextprime, factorint
def A355822_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        k = prod(nextprime(p)**e for p, e in factorint(n).items())
        m, p, c = 1, 2, n
        while c:
            c, a = divmod(c,p)
            m *= p**a
            p = nextprime(p)
        if gcd(k,m) > 1:
            yield n
A355822_list = list(islice(A355822_gen(),30)) # Chai Wah Wu, Jul 18 2022