A368908 - OEIS

%I #26 Jan 13 2024 20:32:20

%S 1,2,3,3,4,5,6,6,7,8,8,8,9,10,10,10,11,12,13,13,13,14,14,14,15,15,15,

%T 15,15,15,15,15,16,17,18,18,18,19,19,19,20,21,22,22,22,22,22,22,23,24,

%U 24,24,24,25,25,25,25,25,26,26,26,26,26,26,27,27,27,27,28,29,29,29,29,29,30,30,30,31,31,31,31,32

%N a(n) is the number of integers x in range 0..n such that the k-th arithmetic derivative of A005940(1+x) is zero for some k >= 0, where A005940 is the Doudna sequence.

%C Doudna-gram for A099308 (numbers whose k-th arithmetic derivative is zero for some k).

%C The creases in the scatter plot graph (these are more easily seen in _Michael De Vlieger_'s attached png image) occur because in general the expected value of A328308(n) decreases if the maximal exponent (A051903) in the prime factorization of n grows, and because the Doudna sequence (A005940) reorders the natural numbers in such a way that numbers with a high value of A051903 occur towards the end of each range [2^n, 2^(n-1)[. The folding effect is even more pronounced in A328307. - _Antti Karttunen_, Jan 12 2024

%H Antti Karttunen, <a href="/A368908/b368908.txt">Table of n, a(n) for n = 0..65537</a>

%H Michael De Vlieger, <a href="/A368908/a368908.png">Scatterplot of a(n)</a>, n = 0..65537 (x axis labeled instead n-1).

%H Michael De Vlieger, <a href="/A368908/a368908_1.png">Log log scatterplot of a(n)</a> n = 0..65537 (x axis labeled instead n-1).

%F a(0) = 1, and for n > 0, a(n) = a(n-1) + A368907(n).

%o (PARI)

%o up_to = 65537;

%o A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };

%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 A328308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n));

%o A368907(n) = A328308(A005940(1+n));

%o A368908list(up_to) = { my(v=vector(up_to), s=A368907(0)); for(i=1, up_to, s += A368907(i); v[i] = s); (v); };

%o v368908 = A368908list(up_to);

%o A368908(n) = if(!n, A368907(0), v368908[n]);

%Y Partial sums of A368907.

%Y Cf. A003415, A005940, A051903, A099308, A328308.

%Y Cf. also A328307, A368906 (compare the scatter plots).

%K nonn

%O 0,2

%A _Antti Karttunen_, Jan 11 2024