A368918 - OEIS

%I #13 Jan 11 2024 09:39:55

%S 0,1,2,3,4,5,6,7,7,8,9,10,11,12,13,14,14,15,16,17,18,19,20,21,21,22,

%T 23,24,25,26,27,28,29,30,31,32,32,33,34,35,36,37,38,39,39,40,41,42,43,

%U 44,45,46,46,47,48,49,50,51,52,53,54,55,56,57,57,58,59,60,61,62,63,64,64,65,66,67,68,69,70,71,71,72

%N a(n) is the number of integers m in the range 0..n such that the arithmetic derivative of A276086(m) has no divisors of the form p^p.

%H Antti Karttunen, <a href="/A368918/b368918.txt">Table of n, a(n) for n = 0..65537</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%F a(0) = 0, for n > 0, a(n) = a(n-1) + A368916(n) = a(n-1) + A359550(A327860(n)).

%F For all n >= 1, a(A368917(n)) = n.

%F For all n >= 0, a(n) >= A328307(n) - 1.

%o (PARI)

%o up_to = 65537;

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

%o A368916(n) = { my(u=A276086(n)); ((u>1)&&A359550(A003415(u))); };

%o A368918list(up_to) = { my(v=vector(up_to), s=A368916(0)); for(i=1, up_to, s +=

%o A368916(i); v[i] = s); (v); };

%o v368918 = A368918list(up_to);

%o A368918(n) = if(!n, 0, v368918[n]);

%Y Partial sums of A368916.

%Y Left inverse of A368917.

%Y Cf. A003415, A276086, A327860, A328307, A359550.

%K nonn

%O 0,3

%A _Antti Karttunen_, Jan 10 2024