cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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).

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

Views

Author

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

Keywords

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
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

Examples

			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 + ...

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.
  • 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

Crossrefs

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.

Programs

  • GAP
    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
  • 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; [n le 1 select 0 else Ad(n): n in [0..80]]; // Bruno Berselli, Oct 22 2013
  • 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]]]; (* 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 *)
  • 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
    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
  • PARI
    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
  • PARI
    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
  • 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):
    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

Formula

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)

Extensions

More terms from Michel ten Voorde, Apr 11 2001

A129150 The n-th arithmetic derivative of 2^3.

Original entry on oeis.org

8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, 8592, 20096, 70464, 235072, 705280, 3023616, 13223680, 55540736, 278539264, 1392697344, 9541095424, 58609614848, 410267320320, 3397142953984, 24143851798528, 176071227916288, 1232666139967488, 9523075842834432
Offset: 0

Views

Author

Reinhard Zumkeller, Apr 01 2007

Keywords

Comments

Conjecture: a strictly increasing sequence. - J. Lowell, Sep 10 2008
The sequence is strictly increasing because (4*n)' = 4*n + 4*n'. - David Radcliffe, Aug 19 2014
8 is the smallest integer that has a nontrivial trajectory (not going to 0 nor reduced to a fixed point as 4) under A003415, but 15 = A090636(1) has 8 as second term in its trajectory. 20 is the next larger such integer with a distinct trajectory, but has two larger predecessors, cf. A090635. - M. F. Hasler, Nov 27 2019
In general, the trajectory of p^(p+1) under A003415 has a common factor p^p, and divided by p^p it gives the trajectory of p under A129283: n -> n + n'. Here we have the case p = 2 (see A129284 for a(n)/2^2), cf. A129151 and A129152 for p = 3 and 5. - M. F. Hasler, Nov 28 2019

Links

Crossrefs

Cf. A129151, A129152, A068327, A099309, A051674, A100716, A129284.
Row n = 8 of A258651.

Programs

  • Haskell
    a129150 n = a129150_list !! n
a129150_list = iterate a003415 8  -- Reinhard Zumkeller, Apr 29 2012
  • Mathematica
    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]]]]]; s = 2^3; Join[{s}, Table[s = dn[s], {28}]] (* T. D. Noe, Mar 07 2013 *)
  • PARI
    A129150(n,a=8)={if(n<0, vector(-n,n, if(n>1, a=A003415(a), a)), for(n=1,n, a=A003415(a)); a)}  \\ For n<0 return the vector a[0..-n-1]. - M. F. Hasler, Nov 27 2019

Formula

a(n+1) = A003415(a(n)), a(0) = 2^3 = 8.
a(n) = A090636(n+2).
A129251(a(n)) > 0. - Reinhard Zumkeller, Apr 07 2007
a(n) = 4*A129284(n). - M. F. Hasler, Nov 27 2019

Extensions

a(21)-a(27) from Paolo P. Lava, Apr 16 2012

A129152 The n-th arithmetic derivative of 5^6.

Original entry on oeis.org

15625, 18750, 34375, 37500, 87500, 187500, 475000, 1212500, 2437500, 6362500, 12737500, 25487500, 50987500, 101987500, 206975000, 530037500, 1060087500, 3890025000, 15175012500, 45525375000, 177026512500, 596222100000, 2708984250000, 12765250350000
Offset: 0

Views

Author

Reinhard Zumkeller, Apr 01 2007

Keywords

Comments

In general, the trajectory of p^(p+1) under A003415 is equal to p^p times the trajectory of p under A129283: n -> n + n'. Here we have the case p = 5 (see A129286 for a(n)/5^5), see A129150 and A129151 for p = 2 and 3. - M. F. Hasler, Nov 28 2019

Crossrefs

Cf. A129150, A129151, A068327.
Cf. A099309, A051674, A100716.

Programs

  • Haskell
    a129152 n = a129152_list !! n
a129152_list = iterate a003415 15625  -- Reinhard Zumkeller, Apr 29 2012
  • Mathematica
    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]]]]]; s = 5^6; Join[{s}, Table[s = dn[s], {18}]] (* T. D. Noe, Mar 07 2013 *)
  • PARI
    A129152_upto(N)=vector(N,n,N=if(n>1,A003415(N),5^6)) \\ gives a(0..N-1). To get a(1..N) put A003415() around if() instead inside.  M. F. Hasler, Nov 28 2019

Formula

a(n+1) = A003415(a(n)), a(0) = 5^6 = 15625.
a(n) = A129286(n)*5^5; A129251(a(n)) > 0. - Reinhard Zumkeller, Apr 07 2007

A129151 The n-th arithmetic derivative of 3^4.

Original entry on oeis.org

81, 108, 216, 540, 1188, 2484, 5076, 10260, 23112, 57996, 135648, 475632, 1586736, 4760640, 20409408, 89259840, 374899968, 1880140032, 9400707072, 64402394112, 395614900224, 2769304412160, 22930714939392, 162970999640064, 1188480788434944, 8320496444780544
Offset: 0

Views

Author

Reinhard Zumkeller, Apr 01 2007

Keywords

Comments

In general, the trajectory of p^(p+1) under A003415 is equal to p^p times the trajectory of p under A129283: n -> n + n'. Here we have the case p = 3 (see A129285 for a(n)/3^3), see A129150 and A129152 for p = 2 and 5. - M. F. Hasler, Nov 28 2019

Crossrefs

Cf. A129150, A129152, A068327.
Cf. A099309, A051674, A100716.

Programs

  • Haskell
    a129151 n = a129151_list !! n
a129151_list = iterate a003415 81  -- Reinhard Zumkeller, Apr 29 2012
  • Mathematica
    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]]]]]; s = 3^4; Join[{s}, Table[s = dn[s], {25}]] (* T. D. Noe, Mar 07 2013 *)

Formula

a(n+1) = A003415(a(n)), a(0) = 3^4 = 81.
a(n) = A129285(n)*3^3; A129251(a(n)) > 0. - Reinhard Zumkeller, Apr 07 2007

A260619 Arithmetic derivative of hyperfactorial(n).

Original entry on oeis.org

0, 0, 4, 216, 165888, 604800000, 48372940800000, 43156963184025600000, 1392410948543163924480000000, 668916177911197542484208831692800000, 8199617664717905359483850194944000000000000000, 2401010998878767104110478543683244630474752000000000000000
Offset: 0

Views

Author

Matthew Campbell, Sep 17 2015

Keywords

Links

Crossrefs

Cf. A002109, A003415, A068327.

Programs

  • Maple
    h:= proc(n) option remember; `if`(n=0, 1, h(n-1)* n^n) end:
a:= proc(n) n^n *`if`(n=0, 0,
      a(n-1)+h(n-1)*n*add(i[2]/i[1], i=ifactors(n)[2]))
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 18 2015
  • Mathematica
    a[n_] := If[n<2, 0, With[{h = Hyperfactorial[n]}, h Sum[{p, e} = pe; e/p, {pe, FactorInteger[h]}]]];
a /@ Range[0, 15] (* Jean-François Alcover, Nov 14 2020 *)

Formula

a(n) = A003415(A002109(n)).
a(n) = A002109(n)*A190121(n) (conjectured).

Extensions

More terms from Alois P. Heinz, Sep 18 2015

A347146 a(n) = Sum_{d|n} (d^d)', where ' is the arithmetic derivative.

Original entry on oeis.org

0, 4, 27, 1028, 3125, 233311, 823543, 201327620, 2324522961, 70000003129, 285311670611, 142657607406431, 302875106592253, 100008061430845691, 3503151123046878152, 590295810358906979332, 827240261886336764177, 826274569581229613840149, 1978419655660313589123979
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 16 2021

Keywords

Comments

Inverse Möbius transform of (n^n)'. - Wesley Ivan Hurt, Mar 31 2025

Examples

			a(4) = (1^1)' + (2^2)' + (4^4)' = 1' + 4' + 256' = 0 + 4 + 1024 = 1028.

Links

Crossrefs

Cf. A003415, A068327.

Programs

  • Maple
    A347146 := proc(n)
    add(A068327(d),d=numtheory[divisors](n)) ;
end proc:
seq( A347146(n),n=1..10) ; # R. J. Mathar, Oct 19 2021
  • Mathematica
    d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#^#] &]; Array[a, 20] (* Amiram Eldar, Oct 16 2021 *)
  • PARI
    ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, ad(d^d)); \\ Michel Marcus, Oct 18 2021

Formula

a(n) = Sum_{d|n} A068327(d). - R. J. Mathar, Oct 19 2021

A348431 a(n) = (n')^(n'), where ' is the arithmetic derivative of n.

Original entry on oeis.org

1, 1, 1, 1, 256, 1, 3125, 1, 8916100448256, 46656, 823543, 1, 18446744073709551616, 1, 387420489, 16777216, 1461501637330902918203684832716283019655932542976, 1, 5842587018385982521381124421, 1, 1333735776850284124449081472843776, 10000000000, 302875106592253
Offset: 0

Views

Author

Wesley Ivan Hurt, Oct 18 2021

Keywords

Comments

a(p) = 1 for primes p since we have a(p) = (p')^(p') = 1^1 = 1.

Crossrefs

Cf. A000312, A003415, A068327.

Programs

  • Maple
    a:= n-> (t-> t^t)(n*add(i[2]/i[1], i=ifactors(n)[2])):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 20 2021
  • Mathematica
    Array[#^# &@ If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &, 19, 2] (* Michael De Vlieger, Oct 18 2021 *)
  • PARI
    ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = my(d=ad(n)); d^d; \\ Michel Marcus, Oct 19 2021

Formula

a(n) = A000312(A003415(n)).
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