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A368908
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.
2
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, 15, 15, 15, 15, 15, 16, 17, 18, 18, 18, 19, 19, 19, 20, 21, 22, 22, 22, 22, 22, 22, 23, 24, 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
OFFSET
0,2
COMMENTS
Doudna-gram for A099308 (numbers whose k-th arithmetic derivative is zero for some k).
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
LINKS
Michael De Vlieger, Scatterplot of a(n), n = 0..65537 (x axis labeled instead n-1).
Michael De Vlieger, Log log scatterplot of a(n) n = 0..65537 (x axis labeled instead n-1).
FORMULA
a(0) = 1, and for n > 0, a(n) = a(n-1) + A368907(n).
PROG
(PARI)
up_to = 65537;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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));
A328308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n));
A368907(n) = A328308(A005940(1+n));
A368908list(up_to) = { my(v=vector(up_to), s=A368907(0)); for(i=1, up_to, s += A368907(i); v[i] = s); (v); };
v368908 = A368908list(up_to);
A368908(n) = if(!n, A368907(0), v368908[n]);
CROSSREFS
Partial sums of A368907.
Cf. also A328307, A368906 (compare the scatter plots).
Sequence in context: A005229 A091245 A347649 * A100618 A248227 A061288
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 11 2024
STATUS
approved