%I A003415 M3196
%S A003415 0,0,1,1,4,1,5,1,12,6,7,1,16,1,9,8,32,1,21,1,24,10,13,1,44,10,15,27,32,
1,
%T A003415 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,
%U A003415 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' = derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = 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 and p' = 1 for any prime p. - 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 (FrankTAW(AT)Netscape.net),
Nov 07 2006
%C A003415 a(m*p^p) = (m + a(m))*p^p, p prime: a(m*A051674(k))=A129283(m)*A051674(k).
- Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 07 2007
%C A003415 See A131116 and A131117 for record values and where they occur. - Reinhard
Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2007
%D A003415 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A003415 E. J. Barbeau, Remarks on an arithmetic derivative, Canad. Math. Bull.,
4 (1961), 117-122.
%D A003415 E. J. Barbeau, Problem, Canad. Math. Congress Notes, 5 (No. 8, April
1973), 6-7.
%D A003415 A. Buium, Differential characters of Abelian varieties over p-adic fields.
Invent. Math. 122 (1995), no. 2, 309-340.
%D A003415 A. Buium, Geometry of p-jets. Duke Math. J. 82 (1996), no. 2, 349-367.
%D A003415 A. Buium, Arithmetic analogues of derivations. J. Algebra 198 (1997),
no. 1, 290-299.
%D A003415 A. Buium, Differential modular forms. J. Reine Angew. Math. 520 (2000),
95-167.
%D A003415 A. M. Gleason et al., The William Lowell Putnam Mathematical Competition:
Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p.
295.
%H A003415 T. D. Noe, <a href="b003415.txt">Table of n, a(n) for n = 0..10000</a>
%H A003415 A. Buium, <a href="http://www.math.unm.edu/~buium">Home Page</a>
%H A003415 Ivars Peterson, <a href="http://www.sciencenews.org/articles/20040320/
mathtrek.asp">Deriving the Structure of Numbers</a>, Science News,
March 20, 2004.
%H A003415 Victor Ufnarovski and Bo Ahlander, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">How to Differentiate a Number</a>, J. Integer
Seqs., Vol. 6, 2003.
%H A003415 Linda Westrick, <a href="http://web.mit.edu/lwest/www/intmain.pdf">Investigations
of the Number Derivative (pdf)</a>
%F A003415 If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
%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.
%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 a(n)=n*sum(i=1,omega(n),factor(n)[i,2]/factor(n)[i,1]) (Paul D. Hanna)
%t A003415 a[0]=0; a[1]=0; a[n_?Negative] := -a[ -n]; a[n_] := Module[{f=Transpose[FactorInteger[n]]},
If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[a[n], {n, 0, 80}]
%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]))} [From Michael Porter (michael_b_porter(AT)yahoo.com),
Nov 25 2009]
%Y A003415 See A038554 for another definition of the derivative of a number.
%Y A003415 A086134 (least prime factor of n'), A086131 (greatest prime factor of
n'), A068719 (derivative of 2n), A068720 (derivative of n^2), A068721
(derivative of n^3), A001787 (derivative of 2^n), A027471 (derivative
of 3^n), A085708 (derivative of 10^n), A068327 (derivative of n^n)
%Y A003415 A024451 (derivative of p#), A068237 (numerator of derivative of 1/n),
A068238 (denominator of derivative of 1/n), A068328 (derivative of
square-free numbers), A068311 (derivative of n!), A068312 (derivative
of triangular numbers), A068329 (derivative of Fibonacci(n))
%Y A003415 A096371 (derivative of partition number) A099301 (derivative of d(n)),
A099310 (derivative of phi(n)) A068346 (second derivative of n),
A099306 (third derivative of n), A085731 (gcd(n, n')), A098699 (least
x such that x' = n),
%Y A003415 A098700 (n such that x' = n has no integer solution), A099302 (number
of solutions to x' = n), A099303 (greatest x such that x' = n), A051674
(n such that n' = n), A099304 (least such that (n+k)' = n' + k'),
A099305 (number of solutions to (n+k)' = n' + k')
%Y A003415 A099307 (least k such that the k-th arithmetic derivative of n is zero),
A099308 (k-th arithmetic derivative of n is zero for some k), A099309
(k-th arithmetic derivative of n is nonzero for all k)
%Y A003415 Cf. A129150, A129151, A129152.
%Y A003415 Sequence in context: A101322 A029644 A024919 this_sequence A086300 A028271
A029666
%Y A003415 Adjacent sequences: A003412 A003413 A003414 this_sequence A003416 A003417
A003418
%K A003415 nonn,easy,nice,new
%O A003415 0,5
%A A003415 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A003415 More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) Apr 11 2001
%E A003415 Additional comments from T. D. Noe (noe(AT)sspectra.com), Oct 12 2004
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