A099302 -id:A099302 - cp's OEIS frontend

A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = ma(n) + n*a(m).

0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, 15, 27, 32, 1, 31, 1, 80, 14, 19, 12, 60, 1, 21, 16, 68, 1, 41, 1, 48, 39, 25, 1, 112, 14, 45, 20, 56, 1, 81, 16, 92, 22, 31, 1, 92, 1, 33, 51, 192, 18, 61, 1, 72, 26, 59, 1, 156, 1, 39, 55, 80, 18, 71

Offset: 0

N. J. A. Sloane, R. K. Guy

Can be extended to negative numbers by defining a(-n) = -a(n).
Based on the product rule for differentiation of functions: for functions f(x) and g(x), (fg)' = f'g + fg'. So with numbers, (ab)' = a'b + ab'. This implies 1' = 0. - Kerry Mitchell, Mar 18 2004
The derivative of a number x with respect to a prime number p as being the number "dx/dp" = (x-x^p)/p, which is an integer due to Fermat's little theorem. - Alexandru Buium, Mar 18 2004
The relation (ab)' = a'b + ab' implies 1' = 0, but it does not imply p' = 1 for p a prime. In fact, any function f defined on the primes can be extended uniquely to a function on the integers satisfying this relation: f(Product_i p_i^e_i) = (Product_i p_i^e_i) * (Sum_i e_i*f(p_i)/p_i). - Franklin T. Adams-Watters, Nov 07 2006
See A131116 and A131117 for record values and where they occur. - Reinhard Zumkeller, Jun 17 2007
Let n be the product of a multiset P of k primes. Consider the k-dimensional box whose edges are the elements of P. Then the (k-1)-dimensional surface of this box is 2*a(n). For example, 2*a(25) = 20, the perimeter of a 5 X 5 square. Similarly, 2*a(18) = 42, the surface area of a 2 X 3 X 3 box. - David W. Wilson, Mar 11 2011
The arithmetic derivative n' was introduced, probably for the first time, by the Spanish mathematician José Mingot Shelly in June 1911 with "Una cuestión de la teoría de los números", work presented at the "Tercer Congreso Nacional para el Progreso de las Ciencias, Granada", cf. link to the abstract on Zentralblatt MATH, and L. E. Dickson, History of the Theory of Numbers. - Giorgio Balzarotti, Oct 19 2013
a(A235991(n)) odd; a(A235992(n)) even. - Reinhard Zumkeller, Mar 11 2014
Sequence A157037 lists numbers with prime arithmetic derivative, i.e., indices of primes in this sequence. - M. F. Hasler, Apr 07 2015
Maybe the simplest "natural extension" of the arithmetic derivative, in the spirit of the above remark by Franklin T. Adams-Watters (2006), is the "pi based" version where f(p) = primepi(p), see sequence A258851. When f is chosen to be the identity map (on primes), one gets A066959. - M. F. Hasler, Jul 13 2015
When n is composite, it appears that a(n) has lower bound 2*sqrt(n), with equality when n is the square of a prime, and a(n) has upper bound (n/2)*log_2(n), with equality when n is a power of 2. - Daniel Forgues, Jun 22 2016
If n = p1*p2*p3*... where p1, p2, p3, ... are all the prime factors of n (not necessarily distinct), and h is a real number (we assume h nonnegative and < 1), the arithmetic derivative of n is equivalent to n' = lim_{h->0} ((p1+h)*(p2+h)*(p3+h)*... - (p1*p2*p3*...))/h. It also follows that the arithmetic derivative of a prime is 1. We could assume h = 1/N, where N is an integer; then the limit becomes {N -> oo}. Note that n = 1 is not a prime and plays the role of constant. - Giorgio Balzarotti, May 01 2023

			6' = (2*3)' = 2'*3 + 2*3' = 1*3 + 2*1 = 5.
Note that, for example, 2' + 3' = 1 + 1 = 2, (2+3)' = 5' = 1. So ' is not linear.
G.f. = x^2 + x^3 + 4*x^4 + x^5 + 5*x^6 + x^7 + 12*x^8 + 6*x^9 + 7*x^10 + ...

G. Balzarotti, P. P. Lava, La derivata aritmetica, Editore U. Hoepli, Milano, 2013.
E. J. Barbeau, Problem, Canad. Math. Congress Notes, 5 (No. 8, April 1973), 6-7.
L. E. Dickson, History of the Theory of Numbers, Vol. 1, Chapter XIX, p. 451, Dover Edition, 2005. (Work originally published in 1919.)
A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p. 295.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Krassimir T. Atanassov, A formula for the n-th prime number, Comptes rendus de l'Académie bulgare des Sciences, Tome 66, No 4, 2013.
E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull. vol. 4, no. 2, May 1961.
A. Buium, Home Page
A. Buium, Differential characters of Abelian varieties over p-adic fields, Invent. Math. 122 (1995), no. 2, 309-340.
A. Buium, Geometry of p-jets, Duke Math. J. 82 (1996), no. 2, 349-367.
A. Buium, Arithmetic analogues of derivations, J. Algebra 198 (1997), no. 1, 290-299.
A. Buium, Differential modular forms, J. Reine Angew. Math. 520 (2000), 95-167.
Brad Emmons and Xiao Xiao, The Arithmetic Partial Derivative, arXiv:2201.12453 [math.NT], 2022.
José María Grau and Antonio M. Oller-Marcén, Giuga Numbers and the Arithmetic Derivative, Journal of Integer Sequences, Vol. 15 (2012), #12.4.1.
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
P. Haukkanen, M. Mattila, J. K. Merikoski and T. Tossavainen, Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?, Journal of Integer Sequences, 16 (2013), #13.1.2. - From _N. J. A. Sloane_, Feb 03 2013
P. Haukkanen, J. K. Merikoski and T. Tossavainen, Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative, Mathematical Communications 25 (2020), 107-115.
Antti Karttunen, Program in LODA-assembly
J. Kovič, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences 15 (2012), Article 12.3.8.
Michael Penn, When the derivative of a number is not zero -- The arithmetic derivative., YouTube video, 2022.
Ivars Peterson, Deriving the Structure of Numbers, Science News, March 20, 2004.
D. J. M. Shelly, Una cuestión de la teoria de los numeros, Asociation Esp. Granada 1911, 1-12 S (1911). (Abstract of ref. JFM42.0209.02 on zbMATH.org)
T. Tossavainen, P. Haukkanen, J. K. Merikoski, and M. Mattila, We can differentiate numbers, too, The College Mathematics Journal 55 (2024), no. 2, 100-108.
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Linda Westrick, Investigations of the Number Derivative, Siemens Foundation competition 2003 and Intel Science Talent Search 2004.
Wikipedia, Arithmetic derivative

Cf. A086134 (least prime factor of n').
Cf. A086131 (greatest prime factor of n').
Cf. A068719 (derivative of 2n).
Cf. A068720 (derivative of n^2).
Cf. A068721 (derivative of n^3).
Cf. A001787 (derivative of 2^n).
Cf. A027471 (derivative of 3^(n-1)).
Cf. A085708 (derivative of 10^n).
Cf. A068327 (derivative of n^n).
Cf. A024451 (derivative of p#).
Cf. A068237 (numerator of derivative of 1/n).
Cf. A068238 (denominator of derivative of 1/n).
Cf. A068328 (derivative of squarefree numbers).
Cf. A068311 (derivative of n!).
Cf. A168386 (derivative of n!!).
Cf. A260619 (derivative of hyperfactorial(n)).
Cf. A260620 (derivative of superfactorial(n)).
Cf. A068312 (derivative of triangular numbers).
Cf. A068329 (derivative of Fibonacci(n)).
Cf. A096371 (derivative of partition number).
Cf. A099301 (derivative of d(n)).
Cf. A099310 (derivative of phi(n)).
Cf. A342925 (derivative of sigma(n)).
Cf. A349905 (derivative of prime shift).
Cf. A327860 (derivative of primorial base exp-function).
Cf. A369252 (derivative of products of three odd primes), A369251 (same sorted).
Cf. A068346 (second  derivative of n).
Cf. A099306 (third   derivative of n).
Cf. A258644 (fourth  derivative of n).
Cf. A258645 (fifth   derivative of n).
Cf. A258646 (sixth   derivative of n).
Cf. A258647 (seventh derivative of n).
Cf. A258648 (eighth  derivative of n).
Cf. A258649 (ninth   derivative of n).
Cf. A258650 (tenth   derivative of n).
Cf. A185232 (n-th    derivative of n).
Cf. A258651 (A(n,k) = k-th arithmetic derivative of n).
Cf. A085731 (gcd(n,n')), A083345 (n'/gcd(n,n')), A057521 (gcd(n, (n')^k) for k>1).
Cf. A342014 (n' mod n), A369049 (n mod n').
Cf. A341998 (A003557(n')), A342001 (n'/A003557(n)).
Cf. A098699 (least x such that x' = n, antiderivative of n).
Cf. A098700 (n such that x' = n has no integer solution).
Cf. A099302 (number of solutions to x' = n).
Cf. A099303 (greatest x such that x' = n).
Cf. A051674 (n such that n' = n).
Cf. A083347 (n such that n' < n).
Cf. A083348 (n such that n' > n).
Cf. A099304 (least k such that (n+k)' = n' + k').
Cf. A099305 (number of solutions to (n+k)' = n' + k').
Cf. A328235 (least k > 0 such that (n+k)' = u * n' for some natural number u).
Cf. A328236 (least m > 1 such that (m*n)' = u * n' for some natural number u).
Cf. A099307 (least k such that the k-th arithmetic derivative of n is zero).
Cf. A099308 (k-th arithmetic derivative of n is zero for some k).
Cf. A099309 (k-th arithmetic derivative of n is nonzero for all k).
Cf. A129150 (n-th derivative of 2^3).
Cf. A129151 (n-th derivative of 3^4).
Cf. A129152 (n-th derivative of 5^6).
Cf. A189481 (x' = n has a unique solution).
Cf. A190121 (partial sums).
Cf. A258057 (first differences).
Cf. A229501 (n divides the n-th partial sum).
Cf. A165560 (parity).
Cf. A235991 (n' is odd), A235992 (n' is even).
Cf. A327863, A327864, A327865 (n' is a multiple of 3, 4, 5).
Cf. A157037 (n' is prime), A192192 (n'' is prime), A328239 (n''' is prime).
Cf. A328393 (n' is squarefree), A328234 (squarefree and > 1).
Cf. A328244 (n'' is squarefree), A328246 (n''' is squarefree).
Cf. A328303 (n' is not squarefree), A328252 (n' is squarefree, but n is not).
Cf. A328248 (least k such that the (k-1)-th derivative of n is squarefree).
Cf. A328251 (k-th arithmetic derivative is never squarefree for any k >= 0).
Cf. A256750 (least k such that the k-th derivative is either 0 or has a factor p^p).
Cf. A327928 (number of distinct primes p such that p^p divides n').
Cf. A342003 (max. exponent k for any prime power p^k that divides n').
Cf. A327929 (n' has at least one divisor of the form p^p).
Cf. A327978 (n' is primorial number > 1).
Cf. A328243 (n' is a partial sum of primorial numbers and larger than one).
Cf. A328310 (maximal prime exponent of n' minus maximal prime exponent of n).
Cf. A328320 (max. prime exponent of n' is less than that of n).
Cf. A328321 (max. prime exponent of n' is >= that of n).
Cf. A328383 (least k such that the k-th derivative of n is either a multiple or a divisor of n, but not both).
Cf. A263111 (the ordinal transform of a).
Cf. A300251, A319684 (Möbius and inverse Möbius transform).
Cf. A305809 (Dirichlet convolution square).
Cf. A349133, A349173, A349394, A349380, A349618, A349619, A349620, A349621 (for miscellaneous Dirichlet convolutions).
Cf. A069359 (similar formula which agrees on squarefree numbers).
Cf. A258851 (the pi-based arithmetic derivative of n).
Cf. A328768, A328769 (primorial-based arithmetic derivatives of n).
Cf. A328845, A328846 (Fibonacci-based arithmetic derivatives of n).
Cf. A302055, A327963, A327965, A328099 (for other variants and modifications).
Cf. A038554 (another sequence using "derivative" in its name, but involving binary expansion of n).
Cf. A322582, A348507 (lower and upper bounds), also A002620.

A003415:= Concatenation([0,0],List(List([2..10^3],Factors),
i->Product(i)*Sum(i,j->1/j))); # Muniru A Asiru, Aug 31 2017
(APL, Dyalog dialect) A003415 ← { ⍺←(0 1 2) ⋄ ⍵≤1:⊃⍺ ⋄ 0=(3⊃⍺)|⍵:((⊃⍺+(2⊃⍺)×(⍵÷3⊃⍺)) ((2⊃⍺)×(3⊃⍺)) (3⊃⍺)) ∇ ⍵÷3⊃⍺ ⋄ ((⊃⍺) (2⊃⍺) (1+(3⊃⍺))) ∇ ⍵} ⍝ Antti Karttunen, Feb 18 2024

a003415 0 = 0
a003415 n = ad n a000040_list where
  ad 1 _             = 0
  ad n ps'@(p:ps)
     | n < p * p     = 1
     | r > 0         = ad n ps
     | otherwise     = n' + p * ad n' ps' where
       (n',r) = divMod n p
-- Reinhard Zumkeller, May 09 2011

Ad:=func; [n le 1 select 0 else Ad(n): n in [0..80]]; // Bruno Berselli, Oct 22 2013

A003415 := proc(n) local B,m,i,t1,t2,t3; B := 1000000000039; if n<=1 then RETURN(0); fi; if isprime(n) then RETURN(1); fi; t1 := ifactor(B*n); m := nops(t1); t2 := 0; for i from 1 to m do t3 := op(i,t1); if nops(t3) = 1 then t2 := t2+1/op(t3); else t2 := t2+op(2,t3)/op(op(1,t3)); fi od: t2 := t2-1/B; n*t2; end;
A003415 := proc(n)
        local a,f;
        a := 0 ;
        for f in ifactors(n)[2] do
                a := a+ op(2,f)/op(1,f);
        end do;
        n*a ;
end proc: # R. J. Mathar, Apr 05 2012

a[ n_] := If[ Abs @ n < 2, 0, n Total[ #2 / #1 & @@@ FactorInteger[ Abs @ n]]]; (* Michael Somos, Apr 12 2011 *)
dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; Table[dn[n], {n, 0, 100}] (* T. D. Noe, Sep 28 2012 *)

A003415(n) = {local(fac);if(n<1,0,fac=factor(n);sum(i=1,matsize(fac)[1],n*fac[i,2]/fac[i,1]))} /* Michael B. Porter, Nov 25 2009 */

apply( A003415(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]), [0..99]) \\ M. F. Hasler, Sep 25 2013, updated Nov 27 2019

A003415(n) = { my(s=0, m=1, spf); while(n>1, spf = A020639(n); n /= spf; s += m*n; m *= spf); (s); }; \\ Antti Karttunen, Mar 10 2021

a(n) = my(f=factor(n), r=[1/(e+!e)|e<-f[,1]], c=f[,2]); n*r*c; \\ Ruud H.G. van Tol, Sep 03 2023

from sympy import factorint
def A003415(n):
    return sum([int(n*e/p) for p,e in factorint(n).items()]) if n > 1 else 0
# Chai Wah Wu, Aug 21 2014

def A003415(n):
    F = [] if n == 0 else factor(n)
    return n * sum(g / f for f, g in F)
[A003415(n) for n in range(79)] # Peter Luschny, Aug 23 2014

If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
a(m*p^p) = (m + a(m))*p^p, p prime: a(m*A051674(k))=A129283(m)*A051674(k). - Reinhard Zumkeller, Apr 07 2007
For n > 1: a(n) = a(A032742(n)) * A020639(n) + A032742(n). - Reinhard Zumkeller, May 09 2011
a(n) = n * Sum_{p|n} v_p(n)/p, where v_p(n) is the largest power of the prime p dividing n. - Wesley Ivan Hurt, Jul 12 2015
For n >= 2, Sum_{k=2..n} floor(1/a(k)) = pi(n) = A000720(n) (see K. T. Atanassov article). - Ivan N. Ianakiev, Mar 22 2019
From A.H.M. Smeets, Jan 17 2020: (Start)
Limit_{n -> oo} (1/n^2)*Sum_{i=1..n} a(i) = A136141/2.
Limit_{n -> oo} (1/n)*Sum_{i=1..n} a(i)/i = A136141.
a(n) = n if and only if n = p^p, where p is a prime number. (End)
Dirichlet g.f.: zeta(s-1)*Sum_{p prime} 1/(p^s-p), see A136141 (s=2), A369632 (s=3) [Haukkanen, Merikoski and Tossavainen]. - Sebastian Karlsson, Nov 25 2021
From Antti Karttunen, Nov 25 2021: (Start)
a(n) = Sum_{d|n} d * A349394(n/d).
For all n >= 1, A322582(n) <= a(n) <= A348507(n).
If n is not a prime, then a(n) >= 2*sqrt(n), or in other words, for all k >= 1 for which A002620(n)+k is not a prime, we have a(A002620(n)+k) > n. [See Ufnarovski and Åhlander, Theorem 9, point (3).]
(End)

More terms from Michel ten Voorde, Apr 11 2001

A369054 Number of representations of n as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1

Offset: 0

Antti Karttunen, Jan 20 2024

nonn

Number of solutions to n = x', where x' is the arithmetic derivative of x (A003415), and x is a product of three odd primes (not all necessarily distinct, A046316).

See the conjecture in A369055.

			a(27) = 1 as 27 can be expressed in exactly one way in the form (p*q + p*r + q*r), with p, q, r all being 3 in this case, as 27 = (3*3 + 3*3 + 3*3).
a(311) = 5 as 311 = (3*5 + 3*37 + 5*37) = (3*7 + 3*29 + 7*29) = (3*13 + 3*17 + 13*17) = (5*7 + 5*23 + 7*23) = (7*11 + 7*13 + 11*13). Expressed in the terms of arithmetic derivatives, of the A099302(311) = 8 antiderivatives of 311 [366, 430, 494, 555, 609, 663, 805, 1001], only the last five are products of three odd primes: 555 = 3*5*37, 609 = 3*7*29, 663 = 3*13*17, 805 = 5*7*23, 1001 = 7 * 11 * 13.

Antti Karttunen, Table of n, a(n) for n = 0..50000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A369055 [quadrisection, a(4n-1)], and its trisections A369460 [= a((12*n)-9)], A369461 [= a((12*n)-5)], A369462 [= a((12*n)-1)].
Cf. A002620, A003415, A046316, A099302, A369056, A369058, A369252.
Cf. A369251 (positions of terms > 0), A369464 (positions of 0's).
Cf. A369063 (positions of records), A369064 (values of records).
Cf. A369241 [= a(2^n - 1)], A369242 [= a(n!-1)], A369245 [= a(A006862(n))], A369247 [= a(3*A057588(n))].

\\ Use this for building up a list up to a certain n. We iterate over weakly increasing triplets of odd primes:
A369054list(up_to) = { my(v = [3,3,3], ip = #v, d, u = vector(up_to)); while(1, d = ((v[1]*v[2]) + (v[1]*v[3]) + (v[2]*v[3])); if(d > up_to, ip--, ip = #v; u[d]++); if(!ip, return(u)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])); };
v369054 = A369054list(100001);
A369054(n) = if(!n,n,v369054[n]);

\\ Use this for computing the value of arbitrary n. We iterate over weakly increasing pairs of odd primes:
A369054(n) = if(3!=(n%4),0, my(v = [3,3], ip = #v, r, c=0); while(1, r = (n-(v[1]*v[2])) / (v[1]+v[2]); if(r < v[2], ip--, ip = #v; if(1==denominator(r) && isprime(r),c++)); if(!ip, return(c)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])));

a(n) = Sum_{i=1..A002620(n)} A369058(i)*[A003415(i)==n], where [ ] is the Iverson bracket.

For n >= 2, a(n) <= A099302(n).

A098700 Numbers n such that x' = n has no integer solution, where x' is the arithmetic derivative of x.

Original entry on oeis.org

2, 3, 11, 17, 23, 29, 35, 37, 47, 53, 57, 65, 67, 79, 83, 89, 93, 97, 107, 117, 125, 127, 137, 145, 149, 157, 163, 173, 177, 179, 189, 197, 205, 207, 209, 217, 219, 223, 233, 237, 245, 257, 261, 277, 289, 303, 305, 307, 317, 323, 325, 337, 345, 353, 367, 373

Offset: 1

Robert G. Wilson v, Sep 21 2004

nonn

If x' = n has solutions, they occur for x <= (n/2)^2. - T. D. Noe, Oct 12 2004
The prime and composite terms are in A189483 and A189554, respectively.
A099302(a(n)) = 0. - Reinhard Zumkeller, Mar 18 2014

T. D. Noe, Table of n, a(n) for n = 1..5000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003. (See p. 7.)

Cf. A003415 (arithmetic derivative of n), A099302 (number of solutions to x' = n), A099303 (greatest x such that x' = n), A098699 (least x such that x' = n).
Cf. A239433 (complement), A002620.
Subsequence of A369464.

a098700 n = a098700_list !! (n-1)
a098700_list = filter
   (\z -> all (/= z) $ map a003415 [1 .. a002620 z]) [2..]
-- Reinhard Zumkeller, Mar 18 2014

a[1] = 0; a[n_] := Block[{f = Transpose[ FactorInteger[ n]]}, If[ PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; b = Table[ -1, {500}]; b[[1]] = 1; Do[c = a[n]; If[c < 500 && b[[c + 1]] == 0, b[[c + 1]] = n], {n, 10^6}]; Select[ Range[500], b[[ # ]] == 0 &]
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}]; Select[Range[400], 0==Count[d1, # ]&]

list(lim)=my(v=List()); lim\=1; forfactored(n=1, lim^2, my(f=n[2],t); listput(v, n[1]*sum(i=1, #f~, f[i, 2]/f[i, 1]))); setminus([1..lim], Set(v)); \\ Charles R Greathouse IV, Oct 21 2021

from itertools import count, islice
from sympy import factorint
def A098700_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda m:sum((m*e//p for p,e in factorint(m).items())) != n,range(1,(n**2>>1)+1))),count(max(startvalue,2)))
A098700_list = list(islice(A098700_gen(),30)) # Chai Wah Wu, Sep 12 2022

Corrected and extended by T. D. Noe, Oct 12 2004

A351029 Number of integers whose arithmetic derivative is equal to the n-th primorial.

Original entry on oeis.org

0, 1, 3, 19, 114, 905, 9494, 124181, 2044847, 43755729, 1043468388, 30309948250

Offset: 1

Antti Karttunen, Feb 01 2022

Number of integers k such that A003415(k) = A002110(n).
a(7) = A116979(7) + 1 since 1547371'=510510 and 1547371=7^2*23*1373 and every other example has only two prime factors. a(8) > A116979(8) because there is at least one term k in A327978 for which A003415(k) = 9699690 = A002110(8), which is not semiprime, that k being 79332523 = 17^2 * 277 * 991. - Edited by Craig J. Beisel, Sep 13 2022 and Antti Karttunen, Jan 05 2023
Most such k are semiprimes, i.e., are "Goldbachian solutions", counted by A116979. The non-semiprime solutions (A366890) form a very tiny minority, and are counted by A369000. - Antti Karttunen, Jan 19 2024

			a(1) = 0 because there are no such k that A003415(k) = 2 = A002110(1).
a(2) = 1 because there is only one number, 9, such that A003415(9) = A002110(2) = 6.
a(3) = 3 because there are exactly three numbers, k = 161, 209, 221, for which A003415(k) = A002110(3) = 30. (See A327978). These are all semiprime solutions, generated by the partitions of 30 into 2 primes: 30 = 7 + 23 = 11 + 19 = 13 + 17, and we have 7*23 = 161; 11*19 = 209; 13*17 = 221.

Cf. A002110, A002620, A003415, A099302, A099303, A116979, A327978, A366890 (nonsemiprime solutions), A368703 (the least of solutions), A368704 (the largest of solutions), A369000.

Cf. also A369239.

A002110(n) = prod(i=1,n,prime(i));
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]));
A351029(n) = { my(g=A002110(n)); sum(k=1,A002620(g),A003415(k)==g); }; \\ Very naive and slow. See comments in A327978.

A351029(n) = {v=prod(j=1,n,prime(j)); c=0; for(k=2, v^2/4, d=0; m=factor(k); for(i=1, matsize(m)[1], d+=(m[i,2]/m[i,1])*k; if(d>v, break;); ); if(d==v, c=c+1; ); ); c;} \\ Craig J. Beisel, Sep 13 2022

a(n) = Sum_{k=1..A002620(A002110(n))} [A003415(k) = A002110(n)], where [ ] is the Iverson bracket.

a(n) = A116979(n) + A369000(n). - Antti Karttunen, Jan 19 2024

a(7) from Craig J. Beisel, Sep 13 2022

a(8)..a(12) [the last based on the value of A116979(12)] from Antti Karttunen, Jan 09 2024

A099303 Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.

Original entry on oeis.org

0, 0, 4, 6, 9, 10, 15, 14, 25, 0, 35, 22, 49, 26, 55, 0, 77, 34, 91, 38, 121, 0, 143, 46, 169, 27, 187, 0, 221, 58, 247, 62, 289, 0, 323, 0, 361, 74, 391, 42, 437, 82, 403, 86, 529, 0, 551, 94, 589, 63, 667, 0, 713, 106, 703, 0, 841, 70, 899, 118, 961, 122, 943, 0, 1073, 0

Offset: 2

T. D. Noe, Oct 12 2004

nonn

This is the largest member of the set I(n) in the paper by Ufnarovski and Ahlander. They show that a(n) <= (n/2)^2.

Because this sequence is quite different for even and odd n, it is bisected into A102084 and A189762. The upper bound for odd n appears to be (n/3)^(3/2), which is attained when n = 3p^2 for primes p>5. - T. D. Noe, Apr 27 2011

See A003415

T. D. Noe, Table of n, a(n) for n=2..1000

Cf. A003415 (arithmetic derivative of n), A099302 (number of solutions to x' = n), A098699 (least x such that x' = n), A098700 (n such that x' = n has no integer solution).

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[x=Max[Flatten[Position[d1, n]]]; If[x>-Infinity, x, 0], {n, 2, 400}]

from sympy import factorint
def A099303(n):
    for m in range(n**2>>2,0,-1):
        if sum((m*e//p for p,e in factorint(m).items())) == n:
            return m
    return 0 # Chai Wah Wu, Sep 12 2022

A189441 Primes that are the arithmetic derivative (A003415) of some number.

Original entry on oeis.org

5, 7, 13, 19, 31, 41, 43, 59, 61, 71, 73, 101, 103, 109, 113, 131, 139, 151, 167, 181, 191, 193, 199, 211, 227, 229, 239, 241, 251, 263, 269, 271, 281, 283, 293, 311, 313, 331, 347, 349, 359, 383, 401, 419, 421, 431, 433, 439, 461, 463, 467, 479, 487, 491

Offset: 1

T. D. Noe, Apr 22 2011

nonn

Complement of A189483 in the primes. Sequence A157037 has the numbers that produce these primes. It is possible for several numbers to produce the same prime. In fact, for each of these primes there is a tree of numbers whose (perhaps multiple) derivatives equal the prime.

Every upper twin prime is here because for such p, the derivative of 2(p-2) is p.

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

Cf. A003415, A099302, A157037.

Primes p such that A099302(p) > 0.

A189483 Primes that are not the arithmetic derivative (A003415) of any integer.

Original entry on oeis.org

2, 3, 11, 17, 23, 29, 37, 47, 53, 67, 79, 83, 89, 97, 107, 127, 137, 149, 157, 163, 173, 179, 197, 223, 233, 257, 277, 307, 317, 337, 353, 367, 373, 379, 389, 397, 409, 443, 449, 457, 499, 509, 547, 569, 577, 593, 613, 659, 673, 677, 683, 709, 733, 757, 769

Offset: 1

T. D. Noe, Apr 22 2011

nonn

These are the prime terms of A098700. Complement of A189441 in the primes.

If we consider rational numbers we can find anti-derivatives for these numbers: for instance 11385/4 = 2846+1/4 is an anti-derivative of 3. - Paolo P. Lava, Aug 02 2012

See A003415.

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

Cf. A003415, A098700, A099302.

Primes p such that A099302(p) = 0.

A327975 Breadth-first reading of the subtree rooted at 5 of the tree where each parent node is the arithmetic derivative (A003415) of all its children.

Original entry on oeis.org

5, 6, 9, 14, 33, 49, 62, 94, 177, 817, 961, 445, 913, 1633, 2173, 2209, 1146, 886, 1822, 4414, 19193, 25829, 32393, 41033, 47429, 57929, 64133, 88229, 101753, 111173, 116729, 129413, 138233, 148553, 160229, 173093, 183929, 188453, 208613, 216773, 232229, 235913, 244229, 249929, 257573, 262793, 272633, 278153, 282533, 288329, 294473, 304613, 316229, 320933, 322853, 323429, 327653, 328313, 1155, 2649

Offset: 1

Antti Karttunen, Oct 02 2019

Permutation of A328115.
The branching degree of vertex v is given by A099302(v).
Leaves form a subsequence of A098700.
Most terms of A189760 (apart from 0, 1, 2, 414, ...) seem to be located in this tree, in positions where they have no smaller siblings.
For any number k at level n (where 5 is at level 2), we have A256750(k) = A327966(k) = n.
Question: Does this subtree contain infinitely long paths? How many? Cf. conjecture number 8 in Ufnarovski and Ahlander paper, and a similar tree starting from 7, A327977.

			Because we have A003415(5) = 1, A003415(6) = 5, A003415(9) = 6, A003415(14) = 9, A003415(33) = A003415(49) = 14, A003415(62) = 33, etc, this subtree is laid out as below. The terms of this sequence are obtained by scanning each successive level of the tree from left to right, from the node 5 onward:
   (0)
    |
   (1)
    |
    5
    |
    6
    |
    9
    |
    14________________
    |                 |
    33               49
    |                 |
    62________       94_____________________________
    |    |    |       |       |      |      |       |
    |    |    |       |       |      |      |       |
   177  817  961     445     913   1633   2173    2209
              |       |       |                     |
              |       |       |                     |
            1146     886    1822                  4414
              |       |       |                     |
              |       |       |                     |
            (19193,  (1155,  (19921, ..., 829921)  (22045, ..., 4870849)
             25829,   2649,                        [49 children for 4414]
               ...,  ...,    [27 children for 1822]
            328313)  196249)
                     [19 children for 886]
        [38 children
         for 1146]
The row lengths thus start as: 1, 1, 1, 1, 2, 2, 8, 4, 133 (= 38+19+27+49), ...

Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A003415, A098699, A098700, A099302, A099303, A099307, A099308, A189760, A256750, A327966, A327968, A328115.

Cf. A327977 for the subtree starting from 7, and also A263267 for another similar tree.

A002620(n) = ((n^2)>>2);
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A327975list(e) = { my(lista=List([5]), f); for(n=1, e, f = lista[n]; for(k=1,1+A002620(f),if(A003415(k)==f, listput(lista,k)))); Vec(lista); };
v328975 = A327975list(21);
A327975(n) = v328975[n];

# uses[A003415]
def A327975():
  '''Breadth-first reading of irregular subtree rooted at 5, defined by the edge-relation A003415(child) = parent.'''
  yield 5
  for x in A327975():
    for k in [1 .. 1+(x*x)//2]:
      if A003415(k) == x: yield k
def take(n, g):
  '''Returns a list composed of the next n elements returned by generator g.'''
  z = []
  if 0 == n: return z
  for x in g:
    z.append(x)
    if n > 1: n = n-1
    else: return(z)
take(60, A327975())

A327977 Breadth-first reading of the subtree rooted at 7 of the tree where each parent node is the arithmetic derivative (A003415) of all its children.

Original entry on oeis.org

7, 10, 21, 25, 18, 38, 46, 65, 77, 217, 361, 129, 205, 493, 529, 98, 426, 718, 170, 254, 462, 982, 1501, 2077, 2257, 2105, 2933, 6953, 11513, 14393, 16469, 17813, 19769, 21653, 24053, 25769, 27413, 29993, 34553, 35369, 41273, 42233, 42869, 44969, 45113, 45173, 11917, 27757, 38881, 45937, 62317, 76897, 84781, 102637, 111457, 114481, 117217, 118477, 120781, 127117, 128881, 501, 1141

Offset: 1

Antti Karttunen, Oct 02 2019

Permutation of A328117.
The branching degree of vertex v is given by A099302(v).
Leaves form a subsequence of A098700.
For any number k at level n (where 7 is at level 2), we have A256750(k) = A327966(k) = n.
Question: Does this subtree contain infinitely long paths? How many? Cf. conjecture number 8 in Ufnarovski and Ahlander paper. As an example of possible beginning of such a sequence they give: 1 ← 7 ← 10 ← 25 ← 46 ← 129 ← 170 ← 501 ← 414 ← 2045.

			The subtree is laid out as below. The terms of this sequence are obtained by scanning each successive level of the tree from left to right, from the node 7 onward:
   (0)
    |
   (1)
    |
    7
    |
    10______________________________
    |                               |
    21________                     25
    |         |                     |
    18___    38_____               46_________________________________
    |    |    |     |               |            |      |             |
    65   77  217   361____         129____      205    493_____      529
         |          |     |         |     |             |      |
         98        426   718       170   254           462    982
         |          |     |         |     |             |      |
        [3]       [21]   [15]      [9]   [9]           [28]   [17]
On the last level illustrated above, the numbers in brackets [ ] tell how many children the node has. E.g, there are three for 98: 1501, 2077, 2257, as A003415(1501) = A003415(2077) = A003415(2257) = 98, and nine for 170: 501, 1141, 2041, 2869, 4309, 5461, 6649, 6901, 7081.

Antti Karttunen, Table of n, a(n) for n = 1..204
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A003415, A098699, A098700, A099302, A099303, A099307, A099308, A189760, A256750, A327966, A327968, A328117.

Cf. A327975 for the subtree starting from 5, and also A263267 for another similar tree.

A002620(n) = ((n^2)>>2);
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A327977list(e) = { my(lista=List([7]), f); for(i=1, e, f = lista[i]; for(k=1,1+A002620(f),if(A003415(k)==f, listput(lista,k)))); Vec(lista); };

\\ With precomputed large A328117, use this:
v328117 = readvec("a328117.txt");
A327977list(e) = { my(lista=List([7]), f, i); for(n=1, e, f = lista[n]; print("n=",n," #lista=", #lista, " A002620(",f,")=",A002620(f)); my(u=1+A002620(f)); if(u>=v328117[#v328117],print("Not enough precomputed terms of A328117 as search upper limit ", u, " > ", v328117[#v328117], " (the last item in v328117). Number of expansions so far=", n); return(1/0)); i=1; while(v328117[i]A003415(v328117[i])==f, listput(lista,v328117[i])); i++)); Vec(lista); };
v327977 = A327977list(114);
A327977(n) = v327977[n];
for(n=1,#v327977,write("b327977.txt", n, " ", A327977(n)));

# uses[A003415]
def A327977():
  '''Breadth-first reading of irregular subtree rooted at 7, defined by the edge-relation A003415(child) = parent. Starts giving terms from 7 onward, after a(0) = 0 and a(1) = 1.'''
  yield 7
  for x in A327977():
    for k in [1 .. 1+floor((x*x)/2)]:
      if(A003415(k) == x): yield k
def take(n, g):
  '''Returns a list composed of the next n elements returned by generator g.'''
  z = []
  if 0 == n: return(z)
  for x in g:
    z.append(x)
    if n > 1: n = n-1
    else: return(z)
take(52, A327977())

cp's OEIS Frontend

A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(m*n) = m*a(n) + n*a(m).

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A369054 Number of representations of n as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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A098700 Numbers n such that x' = n has no integer solution, where x' is the arithmetic derivative of x.

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A351029 Number of integers whose arithmetic derivative is equal to the n-th primorial.

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A099303 Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.

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A189441 Primes that are the arithmetic derivative (A003415) of some number.

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A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = ma(n) + n*a(m).

A369054 Number of representations of n as a sum (pq + pr + q*r) with three odd primes p <= q <= r.