cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

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

Links

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)

A189560 Least odd number k such that x' = k has n solutions, where x' is the arithmetic derivative (A003415) of x.

Original entry on oeis.org

3, 5, 21, 75, 151, 371, 671, 791, 311, 551, 1271, 1391, 1031, 2471, 2231, 4271, 1991, 3191, 5351, 7871, 7751, 7031, 8951, 8711, 11831, 5591, 19631, 10391, 15791, 20711, 30071, 17111, 30551, 27191, 40031, 31391, 52631, 49271, 35591, 42311, 50951, 92231
Offset: 0

Views

Author

T. D. Noe, Apr 24 2011

Keywords

Comments

See A189559 for k restricted to prime numbers and A189558 for no restrictions on k.

References

Links

Crossrefs

Cf. A003415, A189558, A189559.

Programs

  • Python
    from itertools import count
from sympy import factorint
def A189560(n):
    if n == 0:
        return 3
    mdict = {}
    for k in count(1,2):
        c = 0
        for m in range(1,(k**2>>2)+1):
            if m not in mdict:
                mdict[m] = sum((m*e//p for p,e in factorint(m).items()))
            if mdict[m] == k:
                c += 1
            if c > n:
                break
        if c == n:
            return k # Chai Wah Wu, Sep 12 2022

Formula

a(n) is the least odd k such that A099302(k) = n.

A189559 Least prime k such that x' = k has n solutions, where x' is the arithmetic derivative (A003415) of x.

Original entry on oeis.org

2, 5, 31, 167, 151, 491, 983, 887, 311, 1151, 1559, 2111, 1031, 2711, 4391, 4271, 3671, 3191, 5351, 9551, 11471, 14759, 8951, 13751, 11831, 5591, 19991, 10391, 15791, 28031, 30071, 37511, 43151, 27191, 40031, 31391, 52631, 53231, 35591, 52391, 50951
Offset: 0

Views

Author

T. D. Noe, Apr 24 2011

Keywords

Comments

It is much easier finding composite numbers having many solutions. See A189558. For odd numbers, see A189560.

References

Links

Crossrefs

Cf. A003415, A099302, A189558, A189560.

Formula

a(n) is the least prime k such that A099302(k) = n.

A189562 Irregular triangle in which row n has even numbers k that have n solutions to the equation x' = k, where x' denotes the arithmetic derivative (A003415).

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 18, 20, 28, 38, 16, 22, 26, 30, 40, 52, 62, 68, 98, 24, 32, 34, 36, 42, 44, 46, 50, 56, 58, 88, 122, 128, 54, 64, 70, 74, 76, 82, 86, 94, 104, 136, 148, 152, 158, 48, 66, 80, 92, 100, 106, 110, 116, 118, 124, 134, 146, 164, 166, 172, 182
Offset: 0

Views

Author

T. D. Noe, Apr 24 2011

Keywords

Comments

The length of row n is A189561(n). The first term in row n is A189558(n).

Examples

			The triangle begins:
2
4, 6, 8
10, 12, 14, 18, 20, 28, 38
16, 22, 26, 30, 40, 52, 62, 68, 98
24, 32, 34, 36, 42, 44, 46, 50, 56, 58, 88, 122, 128
54, 64, 70, 74, 76, 82, 86, 94, 104, 136, 148, 152, 158
48, 66, 80, 92, 100, 106, 110, 116, 118, 124, 134, 146, 164, 166, 172, 182

References

Links

Crossrefs

Cf. A003415, A099302, A189561.

Formula

Row n has even numbers k such that A099302(k) = n.
Showing 1-4 of 4 results.

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