A355822 - OEIS

%I #16 Jul 21 2022 16:14:58

%S 2,4,6,8,9,10,12,14,15,16,18,20,21,22,24,26,27,28,30,32,34,35,36,38,

%T 39,40,42,44,45,46,48,50,51,52,54,55,56,57,58,60,62,64,65,66,68,69,70,

%U 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

%N 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.

%H Antti Karttunen, <a href="/A355822/b355822.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o A355820(n) = (1==gcd(A003961(n), A276086(n)));

%o isA355822(n) = !A355820(n);

%o (Python)

%o from math import prod, gcd

%o from itertools import count, islice

%o from sympy import nextprime, factorint

%o def A355822_gen(startvalue=1): # generator of terms >= startvalue

%o     for n in count(max(startvalue,1)):

%o         k = prod(nextprime(p)**e for p, e in factorint(n).items())

%o         m, p, c = 1, 2, n

%o         while c:

%o             c, a = divmod(c,p)

%o             m *= p**a

%o             p = nextprime(p)

%o         if gcd(k,m) > 1:

%o             yield n

%o A355822_list = list(islice(A355822_gen(),30)) # _Chai Wah Wu_, Jul 18 2022

%Y Positions of terms > 1 in A355442 and in A355001.

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

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

%Y Cf. also A324584.

%K nonn

%O 1,1

%A _Antti Karttunen_, Jul 18 2022