A300251 -id:A300251 - cp's OEIS frontend

A300252 Difference between arithmetic derivative (A003415) and its Möbius transform (A300251).

0, 0, 0, 1, 0, 2, 0, 4, 1, 2, 0, 8, 0, 2, 2, 12, 0, 10, 0, 10, 2, 2, 0, 24, 1, 2, 6, 12, 0, 17, 0, 32, 2, 2, 2, 32, 0, 2, 2, 32, 0, 21, 0, 16, 13, 2, 0, 64, 1, 16, 2, 18, 0, 42, 2, 40, 2, 2, 0, 56, 0, 2, 15, 80, 2, 29, 0, 22, 2, 25, 0, 88, 0, 2, 17, 24, 2, 33, 0, 88, 27, 2, 0, 72, 2, 2, 2, 56, 0, 73, 2, 28, 2, 2, 2, 160, 0, 22, 19, 62, 0, 41, 0, 64, 27

Offset: 1

Antti Karttunen, Mar 08 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A300245, A300251, A300253.

Cf. A001248 (seems to give the positions of 1's), A006881 (seems to give the positions of 2's).

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A300252(n) = -sumdiv(n,d,(dA003415(d));

a(n) = A003415(n) - A300251(n).

a(n) = -Sum_{d|n, dA008683(n/d)*A003415(d).

A300245 Filter sequence combining arithmetic derivative (A003415) with its Möbius transform (A300251).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 11, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 34, 35, 36, 37, 2, 38, 27, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 40, 60, 61, 62, 2, 63, 36, 64, 65, 66

Offset: 1

Antti Karttunen, Mar 08 2018

nonn

Restricted growth sequence transform of ordered pair [A003415(n), A300251(n)].

			a(16) = a(28) (= 11) because both A003415(16) = A003415(28) = 32 and A300251(16) = A300251(28) = 20.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A300249, A300251, A300252, A300253.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A300251(n) = sumdiv(n,d,moebius(n/d)*A003415(d));
Aux300245(n) = [A003415(n), A300251(n)];
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300245(n))),"b300245.txt");

A300253 GCD of arithmetic derivative (A003415) and its Möbius transform (A300251).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 2, 4, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 4, 1, 1, 1, 16, 2, 1, 2, 4, 1, 1, 2, 4, 1, 1, 1, 16, 13, 1, 1, 16, 1, 1, 2, 2, 1, 3, 2, 4, 2, 1, 1, 4, 1, 1, 3, 16, 2, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 8, 2, 1, 1, 88, 27, 1, 1, 4, 2, 1, 2, 28, 1, 1, 2, 4, 2, 1, 2, 16, 1, 11, 1, 2, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Mar 08 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A085731, A300245, A300251, A300252.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A300251(n) = sumdiv(n,d,moebius(n/d)*A003415(d));
A300253(n) = gcd(A003415(n), A300251(n));

a(n) = gcd(A003415(n), A300251(n)) = gcd(A003415(n), A300252(n)).

A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = ma(n) + n*a(m).

Original entry on oeis.org

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

Offset: 0

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

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 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
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_2(n), with equality when n is a power of 2. - Daniel Forgues, Jun 22 2016
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

			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 + ...

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.
Antti 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)
T. Tossavainen, P. Haukkanen, J. K. Merikoski, and M. Mattila, We can differentiate numbers, too, The College Mathematics Journal 55 (2024), no. 2, 100-108.
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

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-1)).
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 primorial base exp-function).
Cf. A369252 (derivative of products of three odd primes), A369251 (same sorted).
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')), A083345 (n'/gcd(n,n')), A057521 (gcd(n, (n')^k) for k>1).
Cf. A342014 (n' mod n), A369049 (n mod n').
Cf. 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.

A003415:= Concatenation([0,0],List(List([2..10^3],Factors),
i->Product(i)*Sum(i,j->1/j))); # Muniru A Asiru, Aug 31 2017
(APL, Dyalog dialect) A003415 ← { ⍺←(0 1 2) ⋄ ⍵≤1:⊃⍺ ⋄ 0=(3⊃⍺)|⍵:((⊃⍺+(2⊃⍺)×(⍵÷3⊃⍺)) ((2⊃⍺)×(3⊃⍺)) (3⊃⍺)) ∇ ⍵÷3⊃⍺ ⋄ ((⊃⍺) (2⊃⍺) (1+(3⊃⍺))) ∇ ⍵} ⍝ Antti Karttunen, Feb 18 2024

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

Ad:=func; [n le 1 select 0 else Ad(n): n in [0..80]]; // Bruno Berselli, Oct 22 2013

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 *)

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 */

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

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

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

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

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} floor(1/a(k)) = pi(n) = A000720(n) (see K. T. Atanassov article). - Ivan N. Ianakiev, Mar 22 2019
From A.H.M. Smeets, Jan 17 2020: (Start)
Limit_{n -> oo} (1/n^2)*Sum_{i=1..n} a(i) = A136141/2.
Limit_{n -> oo} (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), see A136141 (s=2), A369632 (s=3) [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).]
(End)

More terms from Michel ten Voorde, Apr 11 2001

A349394 a(p^e) = p^(e-1) for prime powers, a(n) = 0 for all other n; Dirichlet convolution of A003415 (arithmetic derivative of n) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 4, 3, 0, 1, 0, 1, 0, 0, 8, 1, 0, 1, 0, 0, 0, 1, 0, 5, 0, 9, 0, 1, 0, 1, 16, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 7, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 32, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 27, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Antti Karttunen, Nov 18 2021

nonn

Dirichlet convolution of this sequence with Euler phi (A000010) is A300251.
Convolving this sequence with sigma (A000203) produces A319684.
With a(1) = 1 instead of 0, this would be the Dirichlet convolution of A129283 (A003415(n)+n) with A055615. Thus when we subtract A063524 from that convolution, we get this sequence. (See also A349434). Compare also to the convolution of A069359 (sequence agreeing with A003415 on squarefree numbers) with A055615, which is the characteristic function of primes, A010051. - Antti Karttunen, Nov 20 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000
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.

Cf. A003415, A003557, A055615, A063524, A069513, A100995, A120007, A129283, A246655.

Cf. also A000010, A000203, A069359, A300251, A319684, A327564, A349340, A349396, A349434, A349618, A349619, A349620, A349621.

import Math.NumberTheory.Primes
a n = case factorise n of
    [(p,e)] -> unPrime p^(e-1) :: Int
     -> 0 -- _Sebastian Karlsson, Nov 19 2021

f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A055615(n) = (n*moebius(n));
A349394(n) = sumdiv(n,d,A003415(n/d)*A055615(d));

A349394(n) = { my(p=0,e); if((e=isprimepower(n,&p)),p^(e-1),0); }; \\ (After Sebastian Karlsson's new formula) - Antti Karttunen, Nov 20 2021

a(n) = Sum_{d|n} A003415(n/d) * A055615(d).
a(n) = 0 unless n is a prime power (A246655), in which case a(p^e) = p^(e-1). - Sebastian Karlsson, Nov 19 2021
a(n) = A003557(n) * A069513(n). [From above] - Antti Karttunen, Nov 20 2021
Dirichlet g.f.: Sum_{p prime} 1/(p^s-p) [Follows from the D.g.f. of A003415 proved by Haukkanen et al.]. - Sebastian Karlsson, Nov 25 2021
Sum_{k=1..n} a(k) has an average value c*n, where c = A137245 = Sum_{primes p} 1/(p*log(p)) = 1.63661632335... - Vaclav Kotesovec, Mar 03 2023

Added Sebastian Karlsson's formula as the new primary definition - Antti Karttunen, Nov 20 2021

A347130 a(n) = Sum_{d|n} d * A003415(n/d), where A003415 is the arithmetic derivative.

Original entry on oeis.org

0, 1, 1, 6, 1, 10, 1, 24, 9, 14, 1, 48, 1, 18, 16, 80, 1, 63, 1, 72, 20, 26, 1, 176, 15, 30, 54, 96, 1, 124, 1, 240, 28, 38, 24, 270, 1, 42, 32, 272, 1, 164, 1, 144, 117, 50, 1, 560, 21, 135, 40, 168, 1, 324, 32, 368, 44, 62, 1, 552, 1, 66, 153, 672, 36, 244, 1, 216, 52, 236, 1, 936, 1, 78, 165, 240, 36, 284, 1, 880

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of the identity function (A000027) with the arithmetic derivative of n (A003415).

Dirichlet convolution of Euler phi (A000010) with A319684.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Inverse Möbius transform of A347131.
Cf. A000010, A000027, A003415, A003557, A319684, A347129.
Cf. also A300251, A305809, A319684, A347132, A347133.

Table[DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &], {n, 80}] (* Michael De Vlieger, Oct 21 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347130(n) = sumdiv(n,d,d*A003415(n/d));

a(n) = Sum_{d|n} d * A003415(n/d).
a(n) = Sum_{d|n} A000010(n/d) * A319684(d).
a(n) = Sum_{d|n} A347131(d).
a(n) = A003557(n) * A347129(n).

A117494 a(n) is the number of m's, 1 <= m <= n, where gcd(m,n) is prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 2, 2, 5, 1, 4, 1, 7, 6, 4, 1, 8, 1, 6, 8, 11, 1, 8, 4, 13, 6, 8, 1, 14, 1, 8, 12, 17, 10, 10, 1, 19, 14, 12, 1, 20, 1, 12, 14, 23, 1, 16, 6, 24, 18, 14, 1, 24, 14, 16, 20, 29, 1, 20, 1, 31, 18, 16, 16, 32, 1, 18, 24, 34, 1, 20, 1, 37, 28, 20, 16, 38, 1, 24, 18, 41, 1, 28

Offset: 1

Leroy Quet, Mar 22 2006

nonn

Dirichlet convolution of A000010 (Euler phi) and A010051 (characteristic function of primes), therefore also Möbius transform of A069359. - Antti Karttunen, Nov 17 2021

			Of the positive integers <= 12, exactly four (2, 3, 9 and 10) have a GCD with 12 that is prime. (gcd(2,12) = 2, gcd(3,12) = 3, gcd(9,12) = 3, gcd(10,12) = 2.)
So a(12) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A008683, A010051, A069359, A085548, A122410, A122411, A175787, A349338.

Coincides with A300251 on squarefree numbers, A005117.

a:=proc(n) local c,m: c:=0: for m from 1 to n do if isprime(gcd(m,n))=true then c:=c+1 else c:=c fi od: end: seq(a(n),n=1..100); # Emeric Deutsch, Apr 01 2006

f[n_] := Length@ Select[GCD[n, Range@n], PrimeQ@ # &]; Array[f, 84] (* Robert G. Wilson v, Apr 06 2006 *)
Table[Count[Range@ n, ?(PrimeQ@ GCD[#, n] &)], {n, 84}] (* _Michael De Vlieger, Feb 25 2018 *)
a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; n * Times @@ (1-1/p) * Total[1/(p - (Boole[# == 1] & /@ e))]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2025 *)

A117494(n) = sum(k=1,n,isprime(gcd(n,k))); \\ Antti Karttunen, Feb 25 2018

a(n) = my(f=factor(n)[, 1]); sum(k=1, #f, eulerphi(n/f[k])); \\ Daniel Suteu, Jun 23 2018

A117494(n) = sumdiv(n,d,eulerphi(n/d)*isprime(d)); \\ Antti Karttunen, Nov 17 2021

Dirichlet g.f: P(s)*Z(s-1)/Z(s) with P(s) the prime zeta function and Z(s) the Riemann zeta function. - Pierre-Louis Giscard, Jul 16 2014
a(n) = Sum_{distinct primes p dividing n} phi(n/p), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 23 2018
From Antti Karttunen, Nov 17 2021: (Start)
a(n) = Sum_{d|n} A008683(n/d) * A069359(d).
a(n) = Sum_{d|n} A000010(n/d) * A010051(d).
a(n) = A349338(n) - A000010(n).
a(A005117(n)) = A300251(A005117(n)) for all n >= 1. (End)
a(n) = 1 iff n = 4 or n is prime (A175787). - Bernard Schott, Nov 18 2021
Sum_{k=1..n} a(k) ~ 3 * A085548 * n^2 / Pi^2. - Vaclav Kotesovec, Nov 20 2021

More terms from Emeric Deutsch, Apr 01 2006

A319684 Sum of A003415(d) over the divisors d of n, where A003415 is arithmetic derivative.

Original entry on oeis.org

0, 1, 1, 5, 1, 7, 1, 17, 7, 9, 1, 27, 1, 11, 10, 49, 1, 34, 1, 37, 12, 15, 1, 83, 11, 17, 34, 47, 1, 54, 1, 129, 16, 21, 14, 114, 1, 23, 18, 117, 1, 68, 1, 67, 55, 27, 1, 227, 15, 64, 22, 77, 1, 142, 18, 151, 24, 33, 1, 190, 1, 35, 69, 321, 20, 96, 1, 97, 28, 90, 1, 326, 1, 41, 75, 107, 20, 110, 1, 325, 142, 45, 1, 244, 24, 47, 34, 219, 1, 243, 22

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Inverse Möbius transform of A003415.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A319683.

Cf. also A300251, A300252, A305809, A317835.

Block[{a}, a[1] = 0; a[n_] := a[n] = If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Array[DivisorSum[#, a[#] &] &, 91]] (* Michael De Vlieger, May 24 2021 *)

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319684(n) = sumdiv(n,d,A003415(d));

a(n) = Sum_{d|n} A003415(d).

a(n) = A319683(n) + A003415(n).

A305809 Dirichlet convolution of A003415 (Arithmetic derivative) with itself.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 8, 1, 2, 0, 18, 0, 2, 2, 40, 0, 22, 0, 22, 2, 2, 0, 96, 1, 2, 12, 26, 0, 40, 0, 160, 2, 2, 2, 147, 0, 2, 2, 128, 0, 48, 0, 34, 28, 2, 0, 400, 1, 34, 2, 38, 0, 156, 2, 160, 2, 2, 0, 276, 0, 2, 32, 560, 2, 64, 0, 46, 2, 56, 0, 680, 0, 2, 36, 50, 2, 72, 0, 560, 90, 2, 0, 348, 2, 2, 2, 224, 0, 346, 2, 58, 2, 2, 2, 1440, 0, 46, 40, 267, 0, 88

Offset: 1

Antti Karttunen, Jun 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A300251, A300252.

Cf. A008578 (positions of 0's), A001248 (of 1's), A006881 (of 2's).

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A305809(n) = sumdiv(n,d,A003415(n/d)*A003415(d));

a(n) = Sum_{d|n} A003415(n/d)*A003415(d).

A347131 a(n) = Sum_{d|n} phi(n/d) * A003415(d), where A003415 is the arithmetic derivative and phi is Euler totient function.

Original entry on oeis.org

0, 1, 1, 5, 1, 8, 1, 18, 8, 12, 1, 33, 1, 16, 14, 56, 1, 45, 1, 53, 18, 24, 1, 110, 14, 28, 45, 73, 1, 87, 1, 160, 26, 36, 22, 169, 1, 40, 30, 182, 1, 119, 1, 113, 93, 48, 1, 328, 20, 107, 38, 133, 1, 216, 30, 254, 42, 60, 1, 337, 1, 64, 125, 432, 34, 183, 1, 173, 50, 183, 1, 538, 1, 76, 135, 193, 34, 215, 1, 552, 216

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A000010 with A003415.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Möbius transform of A347130.
Cf. A000010, A003415.
Cf. also A300251, A305809, A322577, A347132, A347133, A347134, A347235.

f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#] * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Sep 03 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347131(n) = sumdiv(n,d,A003415(n/d)*eulerphi(d));

A347131(n) = sum(k=1,n,A003415(gcd(n,k))); \\ (Slow) - Antti Karttunen, Sep 02 2021

a(n) = Sum_{d|n} A000010(n/d) * A003415(d).
a(n) = Sum_{d|n} A008683(n/d) * A347130(d).
a(n) = Sum_{k=1..n} A003415(gcd(n,k)). - Antti Karttunen, Sep 02 2021

cp's OEIS Frontend

A300252 Difference between arithmetic derivative (A003415) and its Möbius transform (A300251).

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A300245 Filter sequence combining arithmetic derivative (A003415) with its Möbius transform (A300251).

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A300253 GCD of arithmetic derivative (A003415) and its Möbius transform (A300251).

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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).

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A349394 a(p^e) = p^(e-1) for prime powers, a(n) = 0 for all other n; Dirichlet convolution of A003415 (arithmetic derivative of n) with A055615 (Dirichlet inverse of n).

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A347130 a(n) = Sum_{d|n} d * A003415(n/d), where A003415 is the arithmetic derivative.

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A117494 a(n) is the number of m's, 1 <= m <= n, where gcd(m,n) is prime.

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A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = ma(n) + n*a(m).