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A003415
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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)
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181
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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
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OFFSET
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0,5
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COMMENTS
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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 5x5 square. Similarly, 2a(18) = 42, the surface area of a 2x3x3 box. - David W. Wilson, Mar 11 2011
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REFERENCES
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E. J. Barbeau, Remarks on an arithmetic derivative, Canad. Math. Bull., 4 (1961), 117-122.
E. J. Barbeau, Problem, Canad. Math. Congress Notes, 5 (No. 8, April 1973), 6-7.
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.
A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p. 295.
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
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..10000
A. Buium, Home Page
Ivars Peterson, Deriving the Structure of Numbers, Science News, March 20, 2004.
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.
Linda Westrick, Investigations of the Number Derivative (pdf)
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FORMULA
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If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
a(n) = a(A032742(n))*A020639(n) + A032742(n). [Reinhard Zumkeller, May 09 2011]
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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]))} [From Michael B. Porter, Nov 25 2009]
(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
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CROSSREFS
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Cf. A038554 for 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 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).
Sequence in context: A101322 A029644 A024919 * A086300 A028271 A168066
Adjacent sequences: A003412 A003413 A003414 * A003416 A003417 A003418
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, R. K. Guy
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EXTENSIONS
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More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) Apr 11 2001
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STATUS
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approved
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