A328309 a(n) counts the numbers in 0..n whose k-th arithmetic derivative is zero for some k >= 0.

1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 19, 19, 19, 20, 21, 22, 22, 23, 24, 24, 24, 25, 26, 26, 26, 27, 28, 29, 29, 29, 30, 31, 31, 32, 32, 32, 32, 33, 33, 33, 33, 34, 35, 36, 36, 37, 38, 38, 38, 39, 40, 41, 41, 41, 42, 43, 43, 44, 44, 44, 44, 45, 46, 47, 47, 47, 48

Offset: 0

Author

Antti Karttunen, Oct 12 2019

Keywords

• nonn

Comments

a(n) is the number of terms m in A099308 such that m <= n.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537

Crossrefs

Cf. A003415.

Partial sums of A328308. A left inverse of A099308. Cf. also A328307.

Programs

PARI

up_to = 65537;
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));
A328309list(up_to) = { my(v=vector(up_to), s=A328308(0)); for(i=1,up_to,s += A328308(i); v[i] = s); (v); };
v328309 = A328309list(up_to);
A328309(n) = if(!n,1,v328309[n]);

Formula

a(0) = 1; for n > 0, a(n) = a(n-1) + A328308(n).

For all n >= 0, a(A099308(n)) = n.