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A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = m*a(n) + n*a(m).
(Formerly M3196)
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 (list; graph; refs; listen; history; text; internal format)



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 2a(n). For example, 2a(25) = 20, the perimeter of a 5 X 5 square. Similarly, 2a(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 n)/(log 2)), with equality when n is a power of 2. - Daniel Forgues, Jun 22 2016


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.

A. 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)

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


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} [1/a(k)] = pi(n) = A000720(n), where [x] stands for the integer part of x (see K. T. Atanassov article). - Ivan N. Ianakiev, Mar 22 2019

From A.H.M. Smeets, Jan 17 2020: (Start)

Limit_{n -> infinity} (1/n^2)*Sum_{i=1..n} a(i) = A136141/2.

Limit_{n -> infinity} (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) [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).]



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 + ...


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 *)


(PARI) 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 */

(PARI) 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

(PARI) 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


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

(Magma) Ad:=func<h | h*(&+[Factorisation(h)[i][2]/Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [n le 1 select 0 else Ad(n): n in [0..80]]; // Bruno Berselli, Oct 22 2013


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


A003415:= Concatenation([0, 0], List(List([2..10^3], Factors),

i->Product(i)*Sum(i, j->1/j))); # Muniru A Asiru, Aug 31 2017


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).

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 prime product form of primorial base expansion of n).

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')), A057521 (gcd(n, (n')^k) for all k >= 2).

Cf. A342014 (n' mod n), 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.

Sequence in context: A024919 A328385 A328099 * A302055 A086300 A028271

Adjacent sequences:  A003412 A003413 A003414 * A003416 A003417 A003418




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


More terms from Michel ten Voorde, Apr 11 2001



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