A099302 -id:A099302 - cp's OEIS frontend

A189558 Least number k>0 such that x' = k has n solutions, where x' is the arithmetic derivative (A003415) of x.

2, 4, 10, 16, 24, 54, 48, 60, 84, 90, 114, 144, 150, 120, 168, 234, 216, 252, 288, 210, 354, 240, 432, 360, 330, 552, 456, 390, 462, 480, 546, 420, 510, 600, 672, 792, 756, 714, 798, 690, 720, 630, 1116, 1026, 930, 780, 870, 960, 900, 924, 1134, 1296, 840

Offset: 0

T. D. Noe, Apr 24 2011

nonn

It appears that all terms are even. See A189559 for k restricted to prime numbers and A189560 restricted to odd numbers.

See A003415.

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 0..5000 (terms up to a(665) from T. D. Noe)

See A003415, A099302, A189559, A189560.

a(n) is the least k>0 such that A099302(k) = n.

A239433 Numbers m with at least one integer solution for x' = m, where x' is the arithmetic derivative of x (cf. A003415).

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81

Offset: 1

Reinhard Zumkeller, Mar 18 2014

nonn

A099302(a(n)) > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A002620, A098700 (complement), A369251 (subsequence).

a239433 n = a239433_list !! (n-1)
a239433_list = filter
   (\z -> any (== z) $ map a003415 $ takeWhile (<= a002620 z) a013929_list) [2..]

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

T. D. Noe, Apr 24 2011

nonn

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

See A003415.

Donovan Johnson, Table of n, a(n) for n = 0..200

Cf. A003415, A189558, A189559.

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

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

A189553 Irregular triangle in which row n contains numbers x such that x'=n, where x' denotes the arithmetic derivative (A003415).

Original entry on oeis.org

4, 6, 9, 10, 15, 14, 21, 25, 8, 35, 22, 33, 49, 26, 12, 39, 55, 65, 77, 34, 51, 91, 18, 38, 57, 85, 121, 20, 95, 119, 143, 46, 69, 133, 169, 27, 115, 187, 161, 209, 221, 30, 58, 16, 28, 87, 247, 62, 93, 145, 253, 289, 155, 203, 299, 323, 217, 361, 45, 74

Offset: 4

T. D. Noe, Apr 23 2011

Row 0 contains 0 and 1. Row 1 contains all primes. Rows 2 and 3 are empty. Hence, we start this table with row 4. The length of row n is A099302(n). The first term in row n is A098699(n). The last term is A099303(n). Row n is the set I(n) in the paper by Ufnarovski and Ahlander. They show that all terms in row n are <= (n/2)^2. The upper bound is attained when n = 2p, where p is a prime.

			The triangle begins
4
6
9
10
15
14
21, 25
none
8, 35
22
33, 49
26
12, 39, 55

See A003415.

T. D. Noe, Rows n = 4..1000, flattened
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A003415, A098699, A099302, A099303.

dn[0] = 0; dn[1] = 0; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; nn = 100; d = Array[dn, (nn/2)^2]; Table[Flatten[Position[d, n]], {n, 4, nn}]

A189554 Composite numbers n such that x' = n has no integer solution, where x' is the arithmetic derivative (A003415) of x.

Original entry on oeis.org

35, 57, 65, 93, 117, 125, 145, 177, 189, 205, 207, 209, 217, 219, 237, 245, 261, 289, 303, 305, 323, 325, 345, 377, 387, 393, 413, 415, 427, 429, 453, 473, 477, 485, 497, 513, 515, 517, 529, 531, 533, 537, 553, 561, 597, 605, 625, 629, 639, 657, 665, 681

Offset: 1

T. D. Noe, Apr 24 2011

nonn

These are the composite terms of A098700. The prime terms of A098700 are in A189483. Apparently all terms are odd.

See A003415.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A003415, A098700, A099302, A189483.

Composite k such that A099302(k) = 0.

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

T. D. Noe, Apr 24 2011

nonn

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

See A003415.

Donovan Johnson, Table of n, a(n) for n = 0..200

Cf. A003415, A099302, A189558, A189560.

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

A189561 Number of even numbers k that have n solutions to the equation x' = k, where x' denotes the arithmetic derivative (A003415).

Original entry on oeis.org

1, 3, 7, 9, 13, 13, 16, 18, 19, 24, 17, 26, 17, 23, 25, 23, 28, 25, 23, 37, 21, 28, 25, 22, 36, 30, 34, 38, 28, 28, 27, 28, 45, 37, 35, 36, 34, 35, 33, 33, 31, 35, 35, 31, 43, 33, 37, 44, 36, 37, 38, 30, 50, 36, 44, 47, 43, 33, 43, 50, 36, 39, 38, 31, 41, 48

Offset: 0

T. D. Noe, Apr 24 2011

nonn

A189562 lists the even numbers for each n.

See A003415.

T. D. Noe, Table of n, a(n) for n = 0..250

Cf. A003415, A099302, A189562.

a(n) is the number of 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

T. D. Noe, Apr 24 2011

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

			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

See A003415.

T. D. Noe, Rows n = 0..250, flattened

Cf. A003415, A099302, A189561.

Row n has even numbers k such that A099302(k) = n.

A376410 Number of integers whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.

Original entry on oeis.org

0, 1, 4, 13, 40, 186, 952, 5533, 38719, 346207, 3130816, 34444968, 382437431, 4637235152

Offset: 2

Antti Karttunen, Nov 06 2024

For 1! = 1, there are an infinite number of integers k for which A003415(k) = 1 (namely, all the primes), therefore the starting offset is 2.
Like with A351029, also here most of the solutions seem to be squarefree semiprimes, counted by A062311.
Terms a(12)..a(15) were obtained by summing the corresponding terms of A062311 and A377986.

Index entries for sequences related to factorial numbers

Cf. A000142, A002620, A003415, A062311, A099302, A377986.
Cf. A011371, A054861, A027868, A054896.
Cf. also A351029, A369239.

\\ Slow program, for computing just a few terms:
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]));
A376410(n) = { my(g=n!); sum(k=1,A002620(g),A003415(k)==g); };

A376410(n) = AntiDeriv(n!);
AntiDeriv(n,startvlen=1,solsfilename="") = { my(v = vector(startvlen,i,2), ip = #v, r, c=0); while(1, r = A003415vrl(v,n); if(0==r, ip--, if(r > 1, c++; if(solsfilename!="", write(solsfilename, r*factorback(v)))); ip = #v); if(0==ip, v = vector(1+#v,i,2); ip = #v; if(A003415vec(v) > n, return(c)), v[ip] = nextprime(1+v[ip]); for(i=1+ip, #v, v[i]=v[i-1]))); };
A003415vec(tv) = { my(n=factorback(tv), s=0, m=1, spf); for(i=1,#tv,spf = tv[i]; n /= spf; s += m*n; m *= spf); (s); }; \\ Compute Arithmetic derivative from the vector of primes.
A003415vrl(pv,lim) = { my(n=factorback(pv), x=lim-n, s=0, m=1, spf, u=n); for(i=1,#pv,spf = pv[i]; u /= spf; s += m*u; m *= spf); if(((x/s)

a(n) = A099302(A000142(n)).
a(n) = Sum_{k=1..A002620(n!)} [A003415(k) = n!], where [ ] is the Iverson bracket.
a(n) = A062311(n) + A377986(n).

A189555 Numbers n such that x' = n has two solutions, where x' is the arithmetic derivative (A003415) of x.

Original entry on oeis.org

10, 12, 14, 18, 20, 21, 28, 31, 38, 39, 45, 55, 61, 71, 81, 87, 101, 103, 111, 119, 123, 129, 131, 147, 183, 185, 199, 211, 213, 215, 241, 243, 255, 269, 291, 297, 299, 327, 339, 343, 351, 355, 359, 361, 363, 381, 395, 399, 401, 411, 421, 433, 439, 471, 493

Offset: 1

T. D. Noe, Apr 24 2011

nonn

Ufnarovski and Ahlander conjecture that this sequence, and any such sequence that has numbers n such that x' = n has k solutions, is infinite. See A098700 and A189481 for the k=0 and 1 cases. It appears that the only even terms here are 10, 12, 14, 18, 20, 28, and 38. The prime terms are in A189556.

See A003415.

T. D. Noe, Table of n, a(n) for n = 1..1000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A003415, A098700 (no solution), A099302, A189481 (1 solution).

n such that A099302(n) = 2.

cp's OEIS Frontend

A189558 Least number k>0 such that x' = k has n solutions, where x' is the arithmetic derivative (A003415) of x.

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A239433 Numbers m with at least one integer solution for x' = m, where x' is the arithmetic derivative of x (cf. A003415).

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

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A189553 Irregular triangle in which row n contains numbers x such that x'=n, where x' denotes the arithmetic derivative (A003415).

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Mathematica

A189554 Composite numbers n such that x' = n has no integer solution, where x' is the arithmetic derivative (A003415) of x.

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

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A189561 Number of even numbers k that have n solutions to the equation x' = k, where x' denotes the arithmetic derivative (A003415).

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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).

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A376410 Number of integers whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.

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