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%I A351071 #28 May 14 2022 22:22:38
%S A351071 1,4,8,44,216,1474,11130,92489
%N A351071 Number of integers x in range A002110(n) .. A002110(1+n)-1 such that the k-th arithmetic derivative of A276086(x) is zero for some k, where A002110(n) is the n-th primorial.
%C A351071 a(n) is the number of terms of A328116 in range A002110(n) .. A002110(1+n)-1.
%C A351071 a(n) is the number of terms in A351255 (and in A099308) whose largest prime factor (A006530) is A000040(1+n).
%C A351071 Ratio a(n) / A061720(n) develops as:
%C A351071 0: 1 / 1 = 1.0
%C A351071 1: 4 / 4 = 1.0
%C A351071 2: 8 / 24 = 0.333...
%C A351071 3: 44 / 180 = 0.244...
%C A351071 4: 216 / 2100 = 0.1029...
%C A351071 5: 1474 / 27720 = 0.05317...
%C A351071 6: 11130 / 480480 = 0.02316...
%C A351071 7: 92489 / 9189180 = 0.01006...
%C A351071 Computing term a(8) would need processing over 213393180 integers whose greatest prime factor is 23, from single A351255(105368) = 23 at start to product (2^1)*(3^2)*(5*4)*(7^6)*(11^10)*(13^12)*(17^16)*(19^18)*(23^22) at the end of the batch [number whose size in binary is 346 bits], and would required factoring integers of comparable size and more (see A351261), that might not all be easily factorable.
%F A351071 a(n) = Sum_{k=A002110(n) .. A002110(1+n)-1} A328306(k).
%F A351071 a(n) = A328307(A002110(1+n)) - A328307(A002110(n)).
%e A351071 There are eight terms [6, 7, 9, 12, 15, 20, 21, 28] that are >= A002110(2) and < A002110(3) in A328116 for which the corresponding terms [5, 10, 30, 25, 150, 375, 750, 5625] in A276086 (and A351255) are all in A099308, therefore a(2) = 8.
%o A351071 (PARI)
%o A351071 \\ Memoization would work quite badly here. (See comments in A351255. In practice sequence A328306 was computed first, up to its term a(9699690). Same data is available in A328116.)
%o A351071 A002110(n) = prod(i=1,n,prime(i));
%o A351071 A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));
%o A351071 A328308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n));
%o A351071 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A351071 A328306(n) = A328308(A276086(n));
%o A351071 A351071(n) = sum(k=A002110(n),A002110(1+n)-1,A328306(k));
%Y A351071 Cf. A002110, A003415, A061720, A099308, A328306, A328307, A328308, A328116, A351067, A351069, A351255, A351261, A351072 (partial sums).
%Y A351071 Cf. also A327969.
%K A351071 nonn,hard,more
%O A351071 0,2
%A A351071 _Antti Karttunen_, Feb 02 2022