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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)
252
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)
OFFSET

0,5

COMMENTS

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

a(m*p^p) = (m + a(m))*p^p, p prime: a(m*A051674(k))=A129283(m)*A051674(k). - Reinhard Zumkeller, Apr 07 2007

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, 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

REFERENCES

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.

A. Buium, Differential modular forms. J. Reine Angew. Math. 520 (2000), 95-167.

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

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.

Jose Maria Grau and Antonio M. Oller-Marcen, Giuga Numbers and the Arithmetic Derivative, Journal of Integer Sequences, Vol. 15 (2012), #12.4.1.

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

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

FORMULA

If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).

For n > 1: a(n) = a(A032742(n)) * A020639(n) + A032742(n). - Reinhard Zumkeller, May 09 2011

EXAMPLE

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.

MAPLE

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

MATHEMATICA

a[ n_] := If[ Abs @ n < 2, 0, n Total[ #2 / #1 & @@@ FactorInteger[ Abs @ n]]]; Array[a, 80, 0] (* 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 *)

PROG

(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) A003415(n, f)=sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]) \\ - M. F. Hasler, Sep 25 2013

(Haskell)

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

(Python)

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

(Sage)

def A003415(n):

    a = 0; F = []

    if n > 0: F = list(factor(n))

    return n*sum(f[1]/f[0] for f in F)

[A003415(n) for n in range(79)] # Peter Luschny, Aug 23 2014

CROSSREFS

Cf. A038554 (another definition of the derivative of a number).

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. 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. A068346 (second derivative of n).

Cf. A099306 (third derivative of n).

Cf. A085731 (gcd(n,n')).

Cf. A098699 (least x such that x' = 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. 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. A229501 (n divides the n-th partial sum).

Cf. A165560 (parity).

Sequence in context: A101322 A029644 A024919 * A086300 A028271 A168066

Adjacent sequences:  A003412 A003413 A003414 * A003416 A003417 A003418

KEYWORD

nonn,easy,nice,hear

AUTHOR

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

EXTENSIONS

More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org), Apr 11 2001

STATUS

approved

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Last modified December 19 05:27 EST 2014. Contains 252177 sequences.