A099302 Number of integer solutions to x' = n, where x' is the arithmetic derivative of x.

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

Offset: 2

Author

T. D. Noe, Oct 12 2004, Apr 24 2011

Keywords

• nonn
• look

Comments

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.

References

• See A003415

Links

• Antti Karttunen, Table of n, a(n) for n = 2..100000 (terms 2..40000 from T. D. Noe)
• Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Crossrefs

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).

Programs

Haskell

a099302 n = length $ filter (== n) $ map a003415 [1 .. a002620 n]
-- Reinhard Zumkeller, Mar 18 2014

Mathematica

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}]

PARI

up_to = 100000; \\ A002620(10^5) = 2500000000
A002620(n) = ((n^2)>>2);
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);
A099302(n) = v099302[n]; \\ Antti Karttunen, Jan 21 2024

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

Formula

a(A098700(n)) = 0; a(A239433(n)) > 0. - Reinhard Zumkeller, Mar 18 2014

From Antti Karttunen, Jan 21 2024: (Start)

a(n) = Sum_{i=1..A002620(n)} [A003415(i)==n], where [ ] is the Iverson bracket.

a(2n) >= A002375(n), a(2n+1) >= A369054(2n+1).

(End)