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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.

%I A355822 #16 Jul 21 2022 16:14:58
%S A355822 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 A355822 39,40,42,44,45,46,48,50,51,52,54,55,56,57,58,60,62,64,65,66,68,69,70,
%U A355822 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 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.
%H A355822 Antti Karttunen, <a href="/A355822/b355822.txt">Table of n, a(n) for n = 1..10000</a>
%H A355822 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A355822 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%o A355822 (PARI)
%o A355822 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A355822 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A355822 A355820(n) = (1==gcd(A003961(n), A276086(n)));
%o A355822 isA355822(n) = !A355820(n);
%o A355822 (Python)
%o A355822 from math import prod, gcd
%o A355822 from itertools import count, islice
%o A355822 from sympy import nextprime, factorint
%o A355822 def A355822_gen(startvalue=1): # generator of terms >= startvalue
%o A355822     for n in count(max(startvalue,1)):
%o A355822         k = prod(nextprime(p)**e for p, e in factorint(n).items())
%o A355822         m, p, c = 1, 2, n
%o A355822         while c:
%o A355822             c, a = divmod(c,p)
%o A355822             m *= p**a
%o A355822             p = nextprime(p)
%o A355822         if gcd(k,m) > 1:
%o A355822             yield n
%o A355822 A355822_list = list(islice(A355822_gen(),30)) # _Chai Wah Wu_, Jul 18 2022
%Y A355822 Positions of terms > 1 in A355442 and in A355001.
%Y A355822 Cf. A003961, A276086, A355002 (subsequence), A355820 (positions of zeros), A355821 (complement), A355835.
%Y A355822 Cf. A005843 (even numbers, apart from 0, is a subsequence).
%Y A355822 Cf. also A324584.
%K A355822 nonn
%O A355822 1,1
%A A355822 _Antti Karttunen_, Jul 18 2022