A328116 - OEIS

%I #15 Feb 12 2022 17:46:39

%S 0,1,2,3,4,5,6,7,9,12,15,20,21,28,30,31,32,33,35,37,38,40,43,46,47,49,

%T 50,60,61,65,67,68,71,73,74,76,79,84,85,87,91,97,98,104,106,112,118,

%U 119,121,129,133,134,151,153,180,183,196,207,210,211,212,213,218,220,221,223,225,226,227,228,229,231,235,239,240

%N Numbers n such that the k-th arithmetic derivative of A276086(n) is zero for some k.

%C Numbers x such that A276086(x) [which is A351255(a(x))] is in A099308.

%H Antti Karttunen, <a href="/A328116/b328116.txt">Table of n, a(n) for n = 1..10000</a>

%H Antti Karttunen, <a href="/A328116/a328116.txt">First 105368 terms (from 0 to 9699690)</a>

%F For all n >= 1, A328307(a(n)) = n.

%o (PARI)

%o 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 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; };

%o isA099308(n) = { while(n>1, n = A003415checked(n)); (n); };

%o isA328116(n) = isA099308(A276086(n));

%Y Cf. A002110 (subsequence), A003415, A099308, A276086, A327969, A328306 (characteristic function), A328307 (its partial sums).

%Y Cf. A351255 [= A276086(a(n))], A351256 [= A328114(a(n))].

%K nonn

%O 1,3

%A _Antti Karttunen_, Oct 08 2019