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

0,2

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

Formula

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

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

Prog

(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]);

Crossrefs

Partial sums of A328306, a left inverse of A328116.

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

Sequence in context: A373195 A102670 A289411 * A327708 A079631 A347623

Adjacent sequences: A328304 A328305 A328306 * A328308 A328309 A328310

Keyword

nonn

Author

Antti Karttunen, Oct 12 2019

Status

approved