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A285101 a(0) = 2, for n > 0, a(n) = a(n-1)*A242378(n,a(n-1)), where A242378(n,a(n-1)) shifts the prime factorization of a(n-1) n primes towards larger primes with A003961.

%I A285101 #25 May 07 2021 10:42:25
%S A285101 2,6,210,3573570,64845819350301990,
%T A285101 28695662573739152697846686144187168109530,
%U A285101 1038300112150956151877699324649731518883355380534272386781875587619359740733888844803014212990
%N A285101 a(0) = 2, for n > 0, a(n) = a(n-1)*A242378(n,a(n-1)), where A242378(n,a(n-1)) shifts the prime factorization of a(n-1) n primes towards larger primes with A003961.
%C A285101 Multiplicative encoding of irregular table A053632 (in style of A007188 and A260443).
%H A285101 Indranil Ghosh, <a href="/A285101/b285101.txt">Table of n, a(n) for n = 0..9</a>
%F A285101 a(0) = 2, for n > 0, a(n) = a(n-1)*A242378(n,a(n-1)).
%F A285101 Other identities. For all n >= 0:
%F A285101 A001222(a(n)) = A000079(n).
%F A285101 A048675(a(n)) = A028362(1+n).
%F A285101 A248663(a(n)) = A068052(n).
%F A285101 A007913(a(n)) = A285102(n).
%o A285101 (PARI)
%o A285101 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A285101 A242378(k,n) = { while(k>0,n = A003961(n); k = k-1); n; };
%o A285101 A285101(n) = { if(0==n,2,A285101(n-1)*A242378(n,A285101(n-1))); };
%o A285101 (Scheme) (definec (A285101 n) (if (zero? n) 2 (* (A285101 (- n 1)) (A242378bi n (A285101 (- n 1)))))) ;; For A242378bi see A242378.
%o A285101 (Python)
%o A285101 from sympy import factorint, prime, primepi
%o A285101 from operator import mul
%o A285101 from functools import reduce
%o A285101 def a003961(n):
%o A285101     f=factorint(n)
%o A285101     return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**f[i] for i in f])
%o A285101 def a242378(k, n):
%o A285101     while k>0:
%o A285101         n=a003961(n)
%o A285101         k-=1
%o A285101     return n
%o A285101 l=[2]
%o A285101 for n in range(1, 7):
%o A285101     x=l[n - 1]
%o A285101     l.append(x*a242378(n, x))
%o A285101 print(l) # _Indranil Ghosh_, Jun 27 2017
%Y A285101 Cf. A001222, A003961, A007913, A028362, A048675, A053632, A068052, A242378, A248663, A285102.
%Y A285101 Cf. also A007188, A260443.
%K A285101 nonn
%O A285101 0,1
%A A285101 _Antti Karttunen_, Apr 15 2017