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A099302
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Number of integer solutions to x' = n, where x' is the arithmetic derivative of x.
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21
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0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 3, 0, 2, 1, 2, 2, 3, 0, 4, 1, 3, 1, 2, 0, 3, 2, 4, 1, 4, 0, 4, 0, 2, 2, 3, 1, 4, 1, 4, 2, 4, 0, 6, 1, 4, 1, 3, 0, 5, 2, 4, 0, 4, 1, 7, 2, 3, 1, 5, 0, 6, 0, 3, 1, 5, 2, 7, 1, 5, 3, 5, 1, 7, 0, 6, 2, 5, 0, 8, 1, 5, 2, 4, 0, 9, 3, 6, 0, 5, 1, 8, 0, 3, 1, 6, 2, 8, 2, 5, 1, 6
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OFFSET
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2,9
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COMMENTS
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This is the i(n) function in the paper by Ufnarovski and Ahlander. Note that a(1) is infinite because all primes satisfy x' = 1. The plot shows the great difference in the number of solutions for even and odd n. Also compare sequence A189558, which gives the least number have n solutions, and A189560, which gives the least such odd number.
It appears that there are a finite number of even numbers having a given number of solutions. This conjecture is explored in A189561 and A189562.
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REFERENCES
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LINKS
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FORMULA
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a(n) = Sum_{i=1..A002620(n)} [A003415(i)==n], where [ ] is the Iverson bracket.
(End)
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MATHEMATICA
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dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; Table[Count[d1, n], {n, 2, 400}]
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PROG
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(Haskell)
a099302 n = length $ filter (== n) $ map a003415 [1 .. a002620 n]
(Python)
from sympy import factorint
def A099302(n): return sum(1 for m in range(1, (n**2>>2)+1) if sum((m*e//p for p, e in factorint(m).items())) == n) # Chai Wah Wu, Sep 12 2022
(PARI)
up_to = 100000; \\ A002620(10^5) = 2500000000
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A099302list(up_to) = { my(d, c, v=vector(up_to)); for(i=1, A002620(up_to), d = A003415(i); if(d>1 && d<=up_to, v[d]++)); (v); };
v099302 = A099302list(up_to);
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CROSSREFS
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Cf. A002620, A003415 (arithmetic derivative of n), A099303 (greatest x such that x' = n), A098699 (least x such that x' = n), A098700 (n such that x' = n has no integer solution), A239433 (n such that x' = n has at least one solution).
Cf. A002375 (a lower bound for even n), A369054 (a lower bound for n of the form 4m+3).
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KEYWORD
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AUTHOR
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STATUS
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approved
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