A328307 a(n) tells how many numbers m there are in range 0..n such that the k-th arithmetic derivative of A276086(m) is zero for some k >= 0.

1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 15, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 25, 25, 26, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 28, 29, 29, 29, 29, 30, 30, 31, 32, 32, 32, 33, 33, 34, 35, 35, 36, 36, 36, 37, 37, 37, 37, 37, 38

Offset: 0

Author

Antti Karttunen, Oct 12 2019

Keywords

• nonn

Comments

a(n) = the number of such terms m in A328116 that m <= n.

Although in principle A276086 moves any n out of the "all hope lost" zone A100716 (where A328308 is always zero), back to its complement A048103, by comparing the ratio of this and A328309 it can be seen (see the Plot2-link in the Links-section) that such a transfer actually lessens the chances that by just iterating A003415 one could reach zero from there. Note also how the effect of the primorial base representation can be clearly seen in the folds and warps of that plot.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• Ratio of A328309(n)/A328307(n) plotted with OEIS Plot2-script

Crossrefs

Partial sums of A328306, a left inverse of A328116.

Cf. A003415, A048103, A100716, A276086, A327969, A328306, A328308, A328309.

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));
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A328308(n) = if(!n,1, while(n>1, n = A003415checked(n)); (n));
A328306(n) = A328308(A276086(n));
A328307list(up_to) = { my(v=vector(up_to), s=A328306(0)); for(i=1,up_to,s += A328306(i); v[i] = s); (v); };
v328307 = A328307list(up_to);
A328307(n) = if(!n,1,v328307[n]);

Formula

a(0) = 1; for n > 0, a(n) = a(n-1) + A328306(n).

For all n >= 0, a(A328116(n)) = n.