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A284262 a(n) = where A284259 for the first time obtains value n (positions of its records).

%I A284262 #29 May 09 2021 09:51:45
%S A284262 1,2,6,105,5005,85085,1616615,37182145,6685349671,247357937827,
%T A284262 10141675450907,436092044389001,20496326086283047,9156001667401012567,
%U A284262 558516101711461766587,37420578814667938361329,2656861095841423623654359,193950859996423924526768207,15322117939717490037614688353,1271735788996551673122019133299
%N A284262 a(n) = where A284259 for the first time obtains value n (positions of its records).
%H A284262 G. C. Greubel, <a href="/A284262/b284262.txt">Table of n, a(n) for n = 0..335</a>
%F A284262 For n > 1, a(n) = Product_{i = A284263(n)+1 .. A284263(n)+n} prime(i); a(0) = 1, a(1) = 2.
%F A284262 a(n) = A242378(A284263(n), A002110(n)) [shift the prime factorization of the n-th primorial A284263(n) steps towards larger primes].
%F A284262 Other identities. For all n >= 0:
%F A284262 A001221(a(n)) = A001222(a(n))  = n.
%F A284262 A284259(a(n)) = n.
%t A284262 A[n_]:= If[n<1, 0, Block[{k=1}, While[Prime[n + k  - 1] > Prime[k]^2, k++]; k - 1]]; a[n_]:=If[n<2, n + 1, Product[Prime[i], {i, A[n] + 1, A[n] + n}]]; Table[a[n], {n, 0, 51}] (* _Indranil Ghosh_, Mar 24 2017 *)
%o A284262 (Scheme) (define (A284262 n) (A242378bi (A284263 n) (A002110 n))) ;; Where A242378bi(k,n) applies prime shift A003961(n) k times. See A242378.
%o A284262 (PARI) A(n) = { my(k=1); if(0==n, 0, while(prime(n+k-1) > (prime(k)^2), k = k+1); (k-1)); };
%o A284262 a(n) = prod(i=A(n) + 1, A(n) + n, prime(i));
%o A284262 for(n=0, 51, print1(a(n),", ")) \\ _Indranil Ghosh_, after _Antti Karttunen_, Mar 24 2017
%o A284262 (Python)
%o A284262 from sympy import prime
%o A284262 from operator import mul
%o A284262 from functools import reduce
%o A284262 def A(n):
%o A284262     if n<1: return 0
%o A284262     k=1
%o A284262     while prime(n + k - 1)>prime(k)**2:k+=1
%o A284262     return k - 1
%o A284262 def a(n): return n + 1 if n<2 else reduce(mul, [prime(i) for i in range(A(n) + 1, A(n) + n + 1)])
%o A284262 print([a(n) for n in range(21)]) # _Indranil Ghosh_, Mar 24 2017
%Y A284262 Cf. A001221, A001222, A002110, A003961, A242378, A284259 (a left inverse), A284263.
%Y A284262 Cf. also A109819.
%K A284262 nonn
%O A284262 0,2
%A A284262 _Antti Karttunen_, Mar 24 2017