A328244 -id:A328244 - cp's OEIS frontend

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

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

A157037 Numbers with prime arithmetic derivative A003415.

Original entry on oeis.org

6, 10, 22, 30, 34, 42, 58, 66, 70, 78, 82, 105, 114, 118, 130, 142, 154, 165, 174, 182, 202, 214, 222, 231, 238, 246, 255, 273, 274, 282, 285, 286, 298, 310, 318, 345, 357, 358, 366, 370, 382, 385, 390, 394, 399, 418, 430, 434, 442, 454, 455, 465, 474, 478

Offset: 1

Reinhard Zumkeller, Feb 22 2009

nonn

Equivalently, solutions to n'' = 1, since n' = 1 iff n is prime. Twice the lesser of the twin primes, 2*A001359 = A108605, are a subsequence. - M. F. Hasler, Apr 07 2015

All terms are squarefree, because if there would be a prime p whose square p^2 would divide n, then A003415(n) = (A003415(p^2) * (n/p^2)) + (p^2 * A003415(n/p^2)) = p*[(2 * (n/p^2)) + (p * A003415(n/p^2))], which certainly is not a prime. - Antti Karttunen, Oct 10 2019

			A003415(42) = A003415(2*3*7) = 2*3+3*7+7*2 = 41 = A000040(13), therefore 42 is a term.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (first 1000 terms from Reinhard Zumkeller)

Cf. A003415, A010051, A038554, A192082, A192189, A192190,  A327978, A328233, A328240, A328384, A328385.
Cf. A189441 (primes produced by these numbers), A241859.
Cf. A192192, A328239 (numbers whose 2nd and numbers whose 3rd arithmetic derivative is prime).
Cf. A108605, A256673 (subsequences).
Subsequence of following sequences: A005117, A099308, A235991, A328234 (A328393), A328244, A328321.

a157037 n = a157037_list !! (n-1)
a157037_list = filter ((== 1) . a010051' . a003415) [1..]
-- Reinhard Zumkeller, Apr 08 2015

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]]]]]; Select[Range[500], dn[dn[#]] == 1 &] (* T. D. Noe, Mar 07 2013 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA157037(n) = isprime(A003415(n)); \\ Antti Karttunen, Oct 19 2019

from itertools import count, islice
from sympy import isprime, factorint
def A157037_gen(): # generator of terms
    return filter(lambda n:isprime(sum(n*e//p for p,e in factorint(n).items())), count(2))
A157037_list = list(islice(A157037_gen(),20)) # Chai Wah Wu, Jun 23 2022

A010051(A003415(a(n))) = 1; A068346(a(n)) = 1; A099306(a(n)) = 0.

A003415(a(n)) = A328385(a(n)) = A241859(n); A327969(a(n)) = 3. - Antti Karttunen, Oct 19 2019

A328234 Numbers whose arithmetic derivative (A003415) is a squarefree number (A005117) > 1.

Original entry on oeis.org

6, 9, 10, 18, 21, 22, 25, 26, 30, 33, 34, 38, 42, 45, 49, 57, 58, 62, 63, 66, 69, 70, 74, 75, 78, 82, 85, 90, 93, 98, 102, 105, 106, 110, 114, 117, 118, 121, 126, 129, 130, 133, 134, 142, 145, 147, 150, 153, 154, 161, 165, 166, 169, 170, 171, 174, 175, 177, 178, 182, 185, 186, 190, 195, 198, 201, 202, 205, 206, 209, 210, 213

Offset: 1

Antti Karttunen, Oct 10 2019

nonn

Sequence A328393 without primes.
No multiples of 4 because this is a subsequence of A048103.
All terms are cubefree, but being a cubefree non-multiple of 4 doesn't guarantee a membership, as for example 99 = 3^2 * 11 has an arithmetic derivative 11*(2*3) + 3^2 = 75 = 5^2 * 3, and thus is not included in this sequence. (See e.g., A328305).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003415, A005117, A068328, A068719, A235991, A327862, A328239, A328242, A328244, A328245, A328246, A328247.
Cf. A328252 (nonsquarefree terms), A157037, A192192, A327978 (other subsequences).
Subsequence of following sequences: A004709, A048103, A328393.
Complement of the union of A000040 and A328303, i.e., complement of A328303, but without primes.
Cf. also A328248, A328250, A328305.

arthD[n_]:=Module[{fi=FactorInteger[n]},n Total[(fi[[;;,2]]/fi[[;;,1]])]]; Select[Range[300],arthD[#]>1&&SquareFreeQ[arthD[#]]&] (* Harvey P. Dale, Dec 01 2024 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328234(n) = { my(u=A003415(n)); (u>1 && issquarefree(u)); };

A192192 Numbers whose second arithmetic derivative (A068346) is prime; Polynomial-like numbers of degree 3.

Original entry on oeis.org

9, 21, 25, 57, 85, 93, 121, 126, 145, 161, 185, 201, 206, 209, 221, 237, 242, 253, 265, 289, 305, 315, 326, 333, 341, 365, 369, 377, 381, 413, 417, 437, 453, 458, 490, 495, 497, 517, 537, 542, 545, 565, 566, 575, 578, 597, 605, 633, 637, 638, 649, 666, 685

Offset: 1

Vladimir Shevelev, Jun 25 2011

nonn

The fourth A003415-iteration of a(n) is the first to be 0.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (the first 1000 terms from T. D. Noe)

Cf. A003415, A038554, A068346, A192082, A192189, A192190, A327969, A328240.
Cf. A157037, A328239 (the first and third derivative is prime).
Subsequence of following sequences: A328234, A328244, A328246.

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]]]]]; Select[Range[1000], dn[dn[dn[#]]] == 1&] (* T. D. Noe, Mar 07 2013 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA192192(n) = isprime(A003415(A003415(n))); \\ Antti Karttunen, Oct 19 2019

For all n, A327969(a(n)) <= 4. - Antti Karttunen, Oct 19 2019

More terms from Olivier Gérard, Jul 04 2011

New primary definition added to the name by Antti Karttunen, Oct 19 2019

A328245 Numbers whose second arithmetic derivative (A068346) is a squarefree number (A005117), but the first derivative (A003415) is not.

Original entry on oeis.org

14, 46, 50, 65, 77, 86, 94, 99, 122, 125, 138, 146, 207, 230, 302, 334, 343, 346, 375, 426, 531, 546, 554, 581, 590, 626, 662, 682, 686, 710, 717, 718, 725, 734, 747, 750, 819, 842, 869, 875, 931, 965, 1002, 1041, 1083, 1130, 1145, 1146, 1166, 1175, 1202, 1241, 1265, 1310, 1331, 1337, 1349, 1375, 1418, 1461, 1466, 1469, 1501, 1529, 1541

Offset: 1

Antti Karttunen, Oct 11 2019

nonn

			For n = 14, its first arithmetic derivative, A003415(14) = 9 = 3^2 is not squarefree, while the second arithmetic derivative, A003415(9) = 6 = 2* 3 is, thus 14 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003415, A005117, A008966, A068346.
Setwise difference A328244 \ A328234.
Cf. A328253 (a subsequence, nonsquarefree terms).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328245(n) = { my(u=A003415(n)); (!issquarefree(u) && issquarefree(A003415(u))); }; \\ issquarefree(0) returns 0 as zero is not considered as a squarefree number.

A328246 Numbers whose third arithmetic derivative (A099306) is a squarefree number (A005117).

Original entry on oeis.org

9, 14, 18, 21, 25, 33, 38, 46, 49, 57, 65, 77, 85, 93, 98, 121, 126, 129, 134, 138, 141, 145, 150, 161, 166, 177, 185, 186, 194, 201, 205, 206, 209, 217, 221, 237, 242, 249, 253, 258, 262, 265, 266, 289, 290, 301, 305, 306, 315, 322, 326, 333, 334, 338, 341, 342, 350, 361, 365, 369, 375, 377, 381, 393, 398, 402, 413, 414

Offset: 1

Antti Karttunen, Oct 11 2019

nonn

Numbers n for which A008966(A003415(A003415(A003415(n)))) = 1.

			For n=9, its first arithmetic derivative is A003415(9) = 6, its second derivative is A003415(6) = 5, and its third derivative is A003415(5) = 1, and 1 is a squarefree number (in A005117), thus 9 is included in this sequence.
For n=14, A003415(14) = 9, A003415(9) = 6, A003415(6) = 5, and 5, like all primes, is a squarefree number, thus 14 is included in this sequence.
For n=49, A003415(49) = 14, A003415(14) = 9, A003415(9) = 6 = 2*3, and 6 is a squarefree number, thus 49 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003415, A005117, A008966, A099306, A328234, A328244, A328245.

Cf. A192192, A328239, A328247 (subsequences).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328246(n) = { my(u=A003415(A003415(A003415(n)))); (u>0 && issquarefree(u)); };

A328247 Numbers whose third arithmetic derivative (A099306) is a squarefree number (A005117), but the second derivative (A068346) is not.

Original entry on oeis.org

33, 49, 98, 129, 141, 194, 205, 249, 301, 306, 445, 481, 493, 529, 549, 553, 589, 615, 681, 741, 746, 913, 917, 946, 949, 962, 973, 993, 1010, 1106, 1273, 1386, 1397, 1417, 1430, 1518, 1561, 1611, 1633, 1761, 1802, 1842, 1849, 1858, 1870, 1946, 1957, 1977, 2030, 2049, 2078, 2105, 2139, 2166, 2170, 2173, 2175, 2209, 2223, 2330

Offset: 1

Antti Karttunen, Oct 11 2019

nonn

			For n=33, its first arithmetic derivative is A003415(33) = 14, its second derivative is A003415(14) = 9 = 3^2 (which is not squarefree) and its third derivative is A003415(9) = 6 = 2*3, which is, thus 33 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003415, A005117, A008966, A068346, A099306, A328234, A328244, A328245.

Setwise difference A328246 \ A328244.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328247(n) = { my(u=A003415(A003415(n))); (!issquarefree(u) && issquarefree(A003415(u))); };

A368702 Numbers whose second arithmetic derivative (A068346) is a primorial number (A002110) > 1.

Original entry on oeis.org

14, 186, 258, 322, 338, 3318, 3962, 5334, 6106, 7674, 8970, 9186, 9978, 10930, 11994, 12154, 12614, 12970, 13218, 13234, 14626, 15226, 15914, 16378, 17122, 18226, 18658, 19058, 19874, 20194, 20962, 21082, 21106, 21218, 44718, 49358, 57346, 58354, 75442, 76162, 81802, 87814, 95114, 102794, 113505, 114918, 130802

Offset: 1

Antti Karttunen, Jan 09 2024

nonn

Numbers k for which A003415(k) is one of the terms of A327978.

Cf. A002110, A003415, A068346, A327978.

Subsequence of A046100 and of A328244.

A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A068346(n) = A003415(A003415(n));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
isA368702flat(n) = { my(u=A068346(n)); ((u>1)&&(1==A276150(u))); };
k=0; for(n=1, A002620(A002110(7)), if(isA368702flat(n), k++; write("b368702.txt", k, " ", n)));

A348425 Squares whose second arithmetic derivative is a square.

Original entry on oeis.org

0, 1, 4, 49, 529, 2209, 6241, 27889, 28561, 35344, 49729, 128881, 192721, 250000, 431649, 528529, 703921, 1181569, 1495729, 1610361, 1868689, 3411409, 4870849, 5755201, 9138529, 11390625, 12250000, 13830961, 13845841, 15737089, 22648081, 25391521, 31618129

Offset: 1

Marius A. Burtea, Oct 18 2021

nonn

For prime numbers of the form p = k^2 - 2 (A028871) the number m = p^2 is a term because m'' = (p^2)'' = (2*p*p')' = (2*p)'= p + 2*p' = p + 2 = k^2.
If m is a term in A028873 then p = m^2 - 3 is prime and k = p^4 is a term. Indeed: k' = 4*p^3 and k'' = 4*p^3 + 12*p^2 = 4*p^2*(p + 3) = 4*p^2*m^2.
If m is a term in A201787  then p = 5*m^2 - 6 is prime and k = p^6 is a term. Indeed: k' = 6*p^5 and k'' = 5*p^5 + 30*p^4 = 5*p^4*(p + 6) = 25*p^4*m^2.

			4'' = 4' = 4 so 4 is a term.
49'' = 14' = 9 so 49 is a term.

Cf. A000290, A003415, A028871, A028873, A068346, A201787, A266890, A192192, A328244.

f:=func; [n:n in [s*s:s in [0.. 5623]] | IsSquare(Floor(f(Floor(f(n)))))];

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 6000]^2, IntegerQ @ Sqrt[d[d[#]]] &] (* Amiram Eldar, Oct 18 2021 *)

ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k, 2]/f[k, 1])); \\ A003415
lista(nn) = {for (n=0, nn, if (issquare(ad(ad(n^2))), print1(n^2, ", ")); ); } \\ Michel Marcus, Oct 30 2021

cp's OEIS Frontend

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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A157037 Numbers with prime arithmetic derivative A003415.

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A328234 Numbers whose arithmetic derivative (A003415) is a squarefree number (A005117) > 1.

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A192192 Numbers whose second arithmetic derivative (A068346) is prime; Polynomial-like numbers of degree 3.

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A328245 Numbers whose second arithmetic derivative (A068346) is a squarefree number (A005117), but the first derivative (A003415) is not.

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A328246 Numbers whose third arithmetic derivative (A099306) is a squarefree number (A005117).

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A328247 Numbers whose third arithmetic derivative (A099306) is a squarefree number (A005117), but the second derivative (A068346) is not.

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