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.
%I A003415 M3196 #368 Jun 08 2025 16:16:36
%S A003415 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,
%T A003415 32,1,31,1,80,14,19,12,60,1,21,16,68,1,41,1,48,39,25,1,112,14,45,20,
%U A003415 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
%N 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).
%C A003415 Can be extended to negative numbers by defining a(-n) = -a(n).
%C A003415 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
%C A003415 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
%C A003415 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
%C A003415 See A131116 and A131117 for record values and where they occur. - _Reinhard Zumkeller_, Jun 17 2007
%C A003415 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
%C A003415 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
%C A003415 a(A235991(n)) odd; a(A235992(n)) even. - _Reinhard Zumkeller_, Mar 11 2014
%C A003415 Sequence A157037 lists numbers with prime arithmetic derivative, i.e., indices of primes in this sequence. - _M. F. Hasler_, Apr 07 2015
%C A003415 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
%C A003415 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
%C A003415 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
%D A003415 G. Balzarotti, P. P. Lava, La derivata aritmetica, Editore U. Hoepli, Milano, 2013.
%D A003415 E. J. Barbeau, Problem, Canad. Math. Congress Notes, 5 (No. 8, April 1973), 6-7.
%D A003415 L. E. Dickson, History of the Theory of Numbers, Vol. 1, Chapter XIX, p. 451, Dover Edition, 2005. (Work originally published in 1919.)
%D A003415 A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p. 295.
%D A003415 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003415 T. D. Noe, <a href="/A003415/b003415.txt">Table of n, a(n) for n = 0..10000</a>
%H A003415 Krassimir T. Atanassov, <a href="http://doi.baspress.com/pdf-journal/Comptes%20Rendus_4_2013/4_2013_4.pdf">A formula for the n-th prime number</a>, Comptes rendus de l'Académie bulgare des Sciences, Tome 66, No 4, 2013.
%H A003415 E. J. Barbeau, <a href="http://dx.doi.org/10.4153/CMB-1961-013-0">Remark on an arithmetic derivative</a>, Canad. Math. Bull. vol. 4, no. 2, May 1961.
%H A003415 A. Buium, <a href="http://www.math.unm.edu/~buium">Home Page</a>
%H A003415 A. Buium, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002113260">Differential characters of Abelian varieties over p-adic fields</a>, Invent. Math. 122 (1995), no. 2, 309-340.
%H A003415 A. Buium, <a href="http://dx.doi.org/10.1215/S0012-7094-96-08216-2">Geometry of p-jets</a>, Duke Math. J. 82 (1996), no. 2, 349-367.
%H A003415 A. Buium, <a href="http://dx.doi.org/10.1006/jabr.1997.7177">Arithmetic analogues of derivations</a>, J. Algebra 198 (1997), no. 1, 290-299.
%H A003415 A. Buium, <a href="http://dx.doi.org/10.1515/crll.2000.024">Differential modular forms</a>, J. Reine Angew. Math. 520 (2000), 95-167.
%H A003415 Brad Emmons and Xiao Xiao, <a href="https://arxiv.org/abs/2201.12453">The Arithmetic Partial Derivative</a>, arXiv:2201.12453 [math.NT], 2022.
%H A003415 José María Grau and Antonio M. Oller-Marcén, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Oller/oller5.html">Giuga Numbers and the Arithmetic Derivative</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.4.1.
%H A003415 R. K. Guy, <a href="/A003271/a003271.pdf">Letter to N. J. A. Sloane, Apr 1975</a>
%H A003415 P. Haukkanen, M. Mattila, J. K. Merikoski and T. Tossavainen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Tossavainen/tossavainen4.html">Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?</a>, Journal of Integer Sequences, 16 (2013), #13.1.2. - From _N. J. A. Sloane_, Feb 03 2013
%H A003415 P. Haukkanen, J. K. Merikoski and T. Tossavainen, <a href="http://www.mathos.unios.hr/mc/index.php/mc/article/view/3206">Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative</a>, Mathematical Communications 25 (2020), 107-115.
%H A003415 Antti Karttunen, <a href="/A003415/a003415.txt">Program in LODA-assembly</a>
%H A003415 J. Kovič, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kovic/kovic4.html">The Arithmetic Derivative and Antiderivative</a>, Journal of Integer Sequences 15 (2012), Article 12.3.8.
%H A003415 Michael Penn, <a href="https://www.youtube.com/watch?v=lh1qbp7LpFU">When the derivative of a number is not zero -- The arithmetic derivative.</a>, YouTube video, 2022.
%H A003415 Ivars Peterson, <a href="https://www.sciencenews.org/article/deriving-structure-numbers">Deriving the Structure of Numbers</a>, Science News, March 20, 2004.
%H A003415 D. J. M. Shelly, <a href="http://zbmath.org/?q=an:42.0209.02">Una cuestión de la teoria de los numeros</a>, Asociation Esp. Granada 1911, 1-12 S (1911). (Abstract of ref. JFM42.0209.02 on zbMATH.org)
%H A003415 T. Tossavainen, P. Haukkanen, J. K. Merikoski, and M. Mattila, <a href="https://doi.org/10.1080/07468342.2023.2268494">We can differentiate numbers, too</a>, The College Mathematics Journal 55 (2024), no. 2, 100-108.
%H A003415 Victor Ufnarovski and Bo Åhlander, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
%H A003415 Linda Westrick, <a href="https://web.archive.org/web/20050426071741/http://web.mit.edu/lwest/www/intmain.pdf">Investigations of the Number Derivative</a>, Siemens Foundation competition 2003 and Intel Science Talent Search 2004.
%H A003415 Wikipedia, <a href="http://en.wikipedia.org/wiki/Arithmetic_derivative">Arithmetic derivative</a>
%F A003415 If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
%F A003415 a(m*p^p) = (m + a(m))*p^p, p prime: a(m*A051674(k))=A129283(m)*A051674(k). - _Reinhard Zumkeller_, Apr 07 2007
%F A003415 For n > 1: a(n) = a(A032742(n)) * A020639(n) + A032742(n). - _Reinhard Zumkeller_, May 09 2011
%F A003415 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
%F A003415 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
%F A003415 From _A.H.M. Smeets_, Jan 17 2020: (Start)
%F A003415 Limit_{n -> oo} (1/n^2)*Sum_{i=1..n} a(i) = A136141/2.
%F A003415 Limit_{n -> oo} (1/n)*Sum_{i=1..n} a(i)/i = A136141.
%F A003415 a(n) = n if and only if n = p^p, where p is a prime number. (End)
%F A003415 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
%F A003415 From _Antti Karttunen_, Nov 25 2021: (Start)
%F A003415 a(n) = Sum_{d|n} d * A349394(n/d).
%F A003415 For all n >= 1, A322582(n) <= a(n) <= A348507(n).
%F A003415 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).]
%F A003415 (End)
%e A003415 6' = (2*3)' = 2'*3 + 2*3' = 1*3 + 2*1 = 5.
%e A003415 Note that, for example, 2' + 3' = 1 + 1 = 2, (2+3)' = 5' = 1. So ' is not linear.
%e A003415 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 + ...
%p A003415 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;
%p A003415 A003415 := proc(n)
%p A003415 local a,f;
%p A003415 a := 0 ;
%p A003415 for f in ifactors(n)[2] do
%p A003415 a := a+ op(2,f)/op(1,f);
%p A003415 end do;
%p A003415 n*a ;
%p A003415 end proc: # _R. J. Mathar_, Apr 05 2012
%t A003415 a[ n_] := If[ Abs @ n < 2, 0, n Total[ #2 / #1 & @@@ FactorInteger[ Abs @ n]]]; (* _Michael Somos_, Apr 12 2011 *)
%t A003415 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 *)
%o A003415 (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 */
%o A003415 (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
%o A003415 (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
%o A003415 (PARI) 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
%o A003415 (Haskell)
%o A003415 a003415 0 = 0
%o A003415 a003415 n = ad n a000040_list where
%o A003415 ad 1 _ = 0
%o A003415 ad n ps'@(p:ps)
%o A003415 | n < p * p = 1
%o A003415 | r > 0 = ad n ps
%o A003415 | otherwise = n' + p * ad n' ps' where
%o A003415 (n',r) = divMod n p
%o A003415 -- _Reinhard Zumkeller_, May 09 2011
%o A003415 (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
%o A003415 (Python)
%o A003415 from sympy import factorint
%o A003415 def A003415(n):
%o A003415 return sum([int(n*e/p) for p,e in factorint(n).items()]) if n > 1 else 0
%o A003415 # _Chai Wah Wu_, Aug 21 2014
%o A003415 (Sage)
%o A003415 def A003415(n):
%o A003415 F = [] if n == 0 else factor(n)
%o A003415 return n * sum(g / f for f, g in F)
%o A003415 [A003415(n) for n in range(79)] # _Peter Luschny_, Aug 23 2014
%o A003415 (GAP)
%o A003415 A003415:= Concatenation([0,0],List(List([2..10^3],Factors),
%o A003415 i->Product(i)*Sum(i,j->1/j))); # _Muniru A Asiru_, Aug 31 2017
%o A003415 (APL, Dyalog dialect) A003415 ← { ⍺←(0 1 2) ⋄ ⍵≤1:⊃⍺ ⋄ 0=(3⊃⍺)|⍵:((⊃⍺+(2⊃⍺)×(⍵÷3⊃⍺)) ((2⊃⍺)×(3⊃⍺)) (3⊃⍺)) ∇ ⍵÷3⊃⍺ ⋄ ((⊃⍺) (2⊃⍺) (1+(3⊃⍺))) ∇ ⍵} ⍝ _Antti Karttunen_, Feb 18 2024
%Y A003415 Cf. A086134 (least prime factor of n').
%Y A003415 Cf. A086131 (greatest prime factor of n').
%Y A003415 Cf. A068719 (derivative of 2n).
%Y A003415 Cf. A068720 (derivative of n^2).
%Y A003415 Cf. A068721 (derivative of n^3).
%Y A003415 Cf. A001787 (derivative of 2^n).
%Y A003415 Cf. A027471 (derivative of 3^(n-1)).
%Y A003415 Cf. A085708 (derivative of 10^n).
%Y A003415 Cf. A068327 (derivative of n^n).
%Y A003415 Cf. A024451 (derivative of p#).
%Y A003415 Cf. A068237 (numerator of derivative of 1/n).
%Y A003415 Cf. A068238 (denominator of derivative of 1/n).
%Y A003415 Cf. A068328 (derivative of squarefree numbers).
%Y A003415 Cf. A068311 (derivative of n!).
%Y A003415 Cf. A168386 (derivative of n!!).
%Y A003415 Cf. A260619 (derivative of hyperfactorial(n)).
%Y A003415 Cf. A260620 (derivative of superfactorial(n)).
%Y A003415 Cf. A068312 (derivative of triangular numbers).
%Y A003415 Cf. A068329 (derivative of Fibonacci(n)).
%Y A003415 Cf. A096371 (derivative of partition number).
%Y A003415 Cf. A099301 (derivative of d(n)).
%Y A003415 Cf. A099310 (derivative of phi(n)).
%Y A003415 Cf. A342925 (derivative of sigma(n)).
%Y A003415 Cf. A349905 (derivative of prime shift).
%Y A003415 Cf. A327860 (derivative of primorial base exp-function).
%Y A003415 Cf. A369252 (derivative of products of three odd primes), A369251 (same sorted).
%Y A003415 Cf. A068346 (second derivative of n).
%Y A003415 Cf. A099306 (third derivative of n).
%Y A003415 Cf. A258644 (fourth derivative of n).
%Y A003415 Cf. A258645 (fifth derivative of n).
%Y A003415 Cf. A258646 (sixth derivative of n).
%Y A003415 Cf. A258647 (seventh derivative of n).
%Y A003415 Cf. A258648 (eighth derivative of n).
%Y A003415 Cf. A258649 (ninth derivative of n).
%Y A003415 Cf. A258650 (tenth derivative of n).
%Y A003415 Cf. A185232 (n-th derivative of n).
%Y A003415 Cf. A258651 (A(n,k) = k-th arithmetic derivative of n).
%Y A003415 Cf. A085731 (gcd(n,n')), A083345 (n'/gcd(n,n')), A057521 (gcd(n, (n')^k) for k>1).
%Y A003415 Cf. A342014 (n' mod n), A369049 (n mod n').
%Y A003415 Cf. A341998 (A003557(n')), A342001 (n'/A003557(n)).
%Y A003415 Cf. A098699 (least x such that x' = n, antiderivative of n).
%Y A003415 Cf. A098700 (n such that x' = n has no integer solution).
%Y A003415 Cf. A099302 (number of solutions to x' = n).
%Y A003415 Cf. A099303 (greatest x such that x' = n).
%Y A003415 Cf. A051674 (n such that n' = n).
%Y A003415 Cf. A083347 (n such that n' < n).
%Y A003415 Cf. A083348 (n such that n' > n).
%Y A003415 Cf. A099304 (least k such that (n+k)' = n' + k').
%Y A003415 Cf. A099305 (number of solutions to (n+k)' = n' + k').
%Y A003415 Cf. A328235 (least k > 0 such that (n+k)' = u * n' for some natural number u).
%Y A003415 Cf. A328236 (least m > 1 such that (m*n)' = u * n' for some natural number u).
%Y A003415 Cf. A099307 (least k such that the k-th arithmetic derivative of n is zero).
%Y A003415 Cf. A099308 (k-th arithmetic derivative of n is zero for some k).
%Y A003415 Cf. A099309 (k-th arithmetic derivative of n is nonzero for all k).
%Y A003415 Cf. A129150 (n-th derivative of 2^3).
%Y A003415 Cf. A129151 (n-th derivative of 3^4).
%Y A003415 Cf. A129152 (n-th derivative of 5^6).
%Y A003415 Cf. A189481 (x' = n has a unique solution).
%Y A003415 Cf. A190121 (partial sums).
%Y A003415 Cf. A258057 (first differences).
%Y A003415 Cf. A229501 (n divides the n-th partial sum).
%Y A003415 Cf. A165560 (parity).
%Y A003415 Cf. A235991 (n' is odd), A235992 (n' is even).
%Y A003415 Cf. A327863, A327864, A327865 (n' is a multiple of 3, 4, 5).
%Y A003415 Cf. A157037 (n' is prime), A192192 (n'' is prime), A328239 (n''' is prime).
%Y A003415 Cf. A328393 (n' is squarefree), A328234 (squarefree and > 1).
%Y A003415 Cf. A328244 (n'' is squarefree), A328246 (n''' is squarefree).
%Y A003415 Cf. A328303 (n' is not squarefree), A328252 (n' is squarefree, but n is not).
%Y A003415 Cf. A328248 (least k such that the (k-1)-th derivative of n is squarefree).
%Y A003415 Cf. A328251 (k-th arithmetic derivative is never squarefree for any k >= 0).
%Y A003415 Cf. A256750 (least k such that the k-th derivative is either 0 or has a factor p^p).
%Y A003415 Cf. A327928 (number of distinct primes p such that p^p divides n').
%Y A003415 Cf. A342003 (max. exponent k for any prime power p^k that divides n').
%Y A003415 Cf. A327929 (n' has at least one divisor of the form p^p).
%Y A003415 Cf. A327978 (n' is primorial number > 1).
%Y A003415 Cf. A328243 (n' is a partial sum of primorial numbers and larger than one).
%Y A003415 Cf. A328310 (maximal prime exponent of n' minus maximal prime exponent of n).
%Y A003415 Cf. A328320 (max. prime exponent of n' is less than that of n).
%Y A003415 Cf. A328321 (max. prime exponent of n' is >= that of n).
%Y A003415 Cf. A328383 (least k such that the k-th derivative of n is either a multiple or a divisor of n, but not both).
%Y A003415 Cf. A263111 (the ordinal transform of a).
%Y A003415 Cf. A300251, A319684 (Möbius and inverse Möbius transform).
%Y A003415 Cf. A305809 (Dirichlet convolution square).
%Y A003415 Cf. A349133, A349173, A349394, A349380, A349618, A349619, A349620, A349621 (for miscellaneous Dirichlet convolutions).
%Y A003415 Cf. A069359 (similar formula which agrees on squarefree numbers).
%Y A003415 Cf. A258851 (the pi-based arithmetic derivative of n).
%Y A003415 Cf. A328768, A328769 (primorial-based arithmetic derivatives of n).
%Y A003415 Cf. A328845, A328846 (Fibonacci-based arithmetic derivatives of n).
%Y A003415 Cf. A302055, A327963, A327965, A328099 (for other variants and modifications).
%Y A003415 Cf. A038554 (another sequence using "derivative" in its name, but involving binary expansion of n).
%Y A003415 Cf. A322582, A348507 (lower and upper bounds), also A002620.
%K A003415 nonn,core,easy,nice,hear,look
%O A003415 0,5
%A A003415 _N. J. A. Sloane_, _R. K. Guy_
%E A003415 More terms from _Michel ten Voorde_, Apr 11 2001