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.

1, 4, 8, 44, 216, 1474, 11130, 92489

Offset: 0

Author

Antti Karttunen, Feb 02 2022

Keywords

• nonn
• hard
• more

Comments

a(n) is the number of terms of A328116 in range A002110(n) .. A002110(1+n)-1.

a(n) is the number of terms in A351255 (and in A099308) whose largest prime factor (A006530) is A000040(1+n).

Ratio a(n) / A061720(n) develops as:

0: 1 / 1 = 1.0

1: 4 / 4 = 1.0

2: 8 / 24 = 0.333...

3: 44 / 180 = 0.244...

4: 216 / 2100 = 0.1029...

5: 1474 / 27720 = 0.05317...

6: 11130 / 480480 = 0.02316...

7: 92489 / 9189180 = 0.01006...

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.

Examples

			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.
		

Crossrefs

Cf. A002110, A003415, A061720, A099308, A328306, A328307, A328308, A328116, A351067, A351069, A351255, A351261, A351072 (partial sums).

Cf. also A327969.

Programs

PARI

\\ 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.)
A002110(n) = prod(i=1,n,prime(i));
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));
A328308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328306(n) = A328308(A276086(n));
A351071(n) = sum(k=A002110(n),A002110(1+n)-1,A328306(k));

Formula

a(n) = Sum_{k=A002110(n) .. A002110(1+n)-1} A328306(k).

a(n) = A328307(A002110(1+n)) - A328307(A002110(n)).