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A368908
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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.
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2
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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a(0) = 1, and for n > 0, a(n) = a(n-1) + A368907(n).
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PROG
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(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));
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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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