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

2,9

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.

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)

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}]

Prog

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

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).

Sequence in context: A281545 A332033 A331981 * A219198 A352513 A025814

Adjacent sequences: A099299 A099300 A099301 * A099303 A099304 A099305

Keyword

nonn,look

Author

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

Status

approved