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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

A024451 a(n) is the numerator of Sum_{i = 1..n} 1/prime(i).

Original entry on oeis.org

0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853
Offset: 0

Views

Author

N. J. A. Sloane, Clark Kimberling

Keywords

Comments

Arithmetic derivative of p#: a(n) = A003415(A002110(n)). - Reinhard Zumkeller, Feb 25 2002
(n-1)-st elementary symmetric functions of first n primes; see Mathematica section. - Clark Kimberling, Dec 29 2011
Denominators of the harmonic mean of the first n primes; A250130 gives the numerators. - Colin Barker, Nov 14 2014
Let Pn(n) = A002110 denote the primorial function. The average number of distinct prime factors <= prime(n) in the natural numbers up to Pn(n) is equal to Sum_{i = 1..n} 1/prime(i). - Jamie Morken, Sep 17 2018
Conjecture: All terms are squarefree numbers. - Nicolas Bělohoubek, Apr 13 2022
The above conjecture would imply that for n > 0, gcd(a(n), A369651(n)) = 1. See corollary 2 on the page 4 of Ufnarovski-Åhlander paper. - Antti Karttunen, Jan 31 2024
Apart from the initial 0, a subsequence of A048103. Proof: For all primes p, when i >= A000720(p), neither p itself nor p^p divides a(i) [implied by Henry Bottomley's Sep 27 2006 formula], but neither does p^p divide a(i) when 0 < i < A000720(p), as then p^p > a(i). See A074107, which gives an upper bound for this sequence. - Antti Karttunen, Nov 19 2024

Examples

			0/1, 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, ...

References

  • S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2.
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28.

Links

Crossrefs

Denominators are A002110.
See also A106830/A034386, A241189/A241190, A241191/A241192, A061015/A061742, A115963/A115964, A250133/A296358, and A096795/A051451, A354417/A354418, A354859/A354860.
Row sums of A077011 and A258566.
Subsequence of A048103 (after the initial 0).
Cf. A003415, A002110, A070826, A143293, A203008, A250130, A327860, A348301, A369651.
Cf. A053144 (a lower bound), A074107 (an upper bound).
Cf. A109628 (indices k where a(k) is prime), A244622 (corresponding primes), A244621 (a(n) mod 12).
Cf. A369972 (k where prime(1+k)|a(k)), A369973 (corresponding primorials), A293457 (corresponding primes), A377992 (antiderivatives of the terms > 1 of this sequence).
Cf. also A223037, A260615, A274070, A327978, A353299, A353534, A356253.

Programs

  • Magma
    [ Numerator(&+[ NthPrime(k)^-1: k in [1..n]]): n in [1..18] ];  // Bruno Berselli, Apr 11 2011
  • Maple
    h:= n-> add(1/(ithprime(i)),i=1..n);
t1:=[seq(h(n),n=0..50)];
t1a:=map(numer,t1); # A024451
t1b:=map(denom,t1); # A002110 - N. J. A. Sloane, Apr 25 2014
  • Mathematica
    a[n_] := Numerator @ Sum[1/Prime[i], {i, n}]; Array[a,18]  (* Jean-François Alcover, Apr 11 2011 *)
f[k_] := Prime[k]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A024451 *)
(* Clark Kimberling, Dec 29 2011 *)
Numerator[Accumulate[1/Prime[Range[20]]]] (* Harvey P. Dale, Apr 11 2012 *)
  • PARI
    a(n) = numerator(sum(i=1, n, 1/prime(i))); \\ Michel Marcus, Sep 18 2018
  • Python
    from sympy import prime
from fractions import Fraction
def a(n): return sum(Fraction(1, prime(k)) for k in range(1, n+1)).numerator
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 12 2021
  • Python
    from math import prod
from sympy import prime
def A024451(n):
    q = prod(plist:=tuple(prime(i) for i in range(1,n+1)))
    return sum(q//p for p in plist) # Chai Wah Wu, Nov 03 2022

Formula

Limit_{n->oo} (Sum_{p <= n} 1/p - log log n) = 0.2614972... = A077761.
a(n) = (Product_{i=1..n} prime(i))*(Sum_{i=1..n} 1/prime(i)). - Benoit Cloitre, Jan 30 2002
(n+1)-st elementary symmetric function of the first n primes.
a(n) = a(n-1)*A000040(n) + A002110(n-1). - Henry Bottomley, Sep 27 2006
From Antti Karttunen, Jan 31 2024, Feb 08 2024 and Nov 19 2024: (Start)
a(0) = 0, for n > 0, a(n) = 2*A203008(n-1) + A070826(n).
For n > 0, a(n) = A327860(A143293(n-1)).
For n > 0, a(n) = A348301(n) + A002110(n).
For n = 3..175, a(n) = A356253(A002110(n)). [See comments in A356253.]
For n >= 0, A053144(n) <= a(n) <= A074107(n) < A070826(1+n).
(End)

Extensions

a(0)=0 prepended by Alois P. Heinz, Jun 26 2015

A327969 The length of a shortest path from n to zero when using the transitions x -> A003415(x) and x -> A276086(x), or -1 if no zero can ever be reached from n.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 3, 2, 6, 4, 3, 2, 5, 2, 5, 6, 6, 2, 5, 2, 7, 4, 3, 2
Offset: 0

Views

Author

Antti Karttunen, Oct 07 2019

Keywords

Comments

The terms of this sequence are currently known only up to n=23, with the value of a(24) still being uncertain. For the tentative values of the later terms, see sequence A328324 which gives upper bounds for these terms, many of which are very likely also exact values for them.
As A051903(A003415(n)) >= A051903(n)-1, it means that it takes always at least A051903(n) steps to a prime if iterating solely with A003415.
Some known values and upper bounds from n=24 onward:
a(24) <= 11.
a(25) = 4.
a(26) = 7.
a(27) <= 22.
a(33) = 4.
a(39) = 4.
a(40) = 5.
a(42) = 3.
a(44) <= 10.
a(45) = 5.
a(46) = 5.
a(48) = 9.
a(49) = 6.
a(50) = 6.
a(55) = 7.
a(74) = 5.
a(77) = 6.
a(80) <= 18.
a(111) = 6.
a(112) = 8.
a(125) <= 9.
a(240) = 7.
a(625) <= 10.
a(875) = 8.
From Antti Karttunen, Feb 20 2022: (Start)
a(2556) <= 20.
a(5005) <= 19.
What is the value of a(128), and is A328324(128) well-defined?
When I created this sequence, I conjectured that by applying two simple arithmetic operations "arithmetic derivative" (A003415) and "primorial base exp-function" (A276086) in some combination, and starting from any positive integer, we could always reach zero (via a prime and 1).
At the first sight it seems almost certain that the conjecture holds, as it is always possible at every step to choose from two options (which very rarely meet, see A351088), leading to an exponentially growing search tree, and also because A276086 always jumps out of any dead-end path with p^p-factors (dead-end from the arithmetic derivative's point of view). However, it should be realized that one can reach the terms of either A157037 or A327978 with a single step of A003415 only from squarefree numbers (or respectively, cubefree numbers that are not multiples of 4, see A328234), and in general, because A003415 decreases the maximal exponent of the prime factorization (A051903) at most by one, if the maximal exponent in the prime factorization of n is large, there is a correspondingly long path to traverse if we take only A003415-steps in the iteration, and any step could always lead with certain probability to a p^p-number. Note that the antiderivatives of primorials with a square factor seem quite rare, see A351029.
And although taking a A276086-step will always land us to a p^p-free number (which a priori is not in the obvious dead-end path of A003415, although of course it might eventually lead to one), it (in most cases) also increases the magnitude of number considerably, that tends to make the escape even harder. Particularly, in the majority of cases A276086 increases the maximal exponent (which in the preimage is A328114, "maximal digit value used when n is written in primorial base"), so there will be even a longer journey down to squarefree numbers when using A003415. See the sequences A351067 and A351071 for the diminishing ratios suggesting rapidly diminishing chances of successfully reaching zero from larger terms of A276086. Also, the asymptotic density of A276156 is zero, even though A351073 may contain a few larger values.
On the other hand, if we could prove that by (for example) continuing upwards with any p^p-path of A003415 we could eventually reach with a near certainty a region of numbers with low values of A328114 (i.e., numbers with smallish digits in primorial base, like A276156), then the situation might change (see also A351089). However, a few empirical runs seemed to indicate otherwise.
For all of the above reasons, I now conjecture that there are natural numbers from which it is not possible to reach zero with any combination of steps. For example 128 or 5^5 = 3125.
(End)

Examples

			Let -A> stand for an application of A003415 and -B> for an application of A276086, then, we have for example:
a(8) = 6 as we have 8 -A>  12 -B>  25 -A> 10 -A>  7 -A> 1 -A> 0, six transitions in total (and there are no shorter paths).
a(15) = 6 as we have 15 -B> 150 -A> 185 -A> 42 -A> 41 -A> 1 -A> 0, six transitions in total (and there are no shorter paths).
a(20) = 7, as 20 -B> 375 -A> 350 -A> 365 -A> 78 -A> 71 -A> 1 -A> 0, and there are no shorter paths.
For n=112, we know that a(112) cannot be larger than eight, as A328099^(8)(112) = 0, so we have a path of length 8 as 112 -A> 240 -B> 77 -A> 18 -A> 21 -A> 10 -A> 7 -A> 1 -A> 0. Checking all 32 combinations of the paths of lengths of 5 starting from 112 shows that none of them or their prefixes ends with a prime, thus there cannot be any shorter path, and indeed a(112) = 8.
a(24) <= 11 as A328099^(11)(24) = 0, i.e., we have 24 -A> 44 -A> 48 -A> 112 -A> 240 -B> 77 -A> 18 -A> 21 -A> 10 -A> 7 -A> 1 -A> 0. On the other hand, 24 -B> 625 -B> 17794411250 -A> 41620434625 -A> 58507928150 -A> 86090357185 -A> 54113940517 -A> 19982203325 -A> 12038411230 -A> 8426887871 -A> 1 -A> 0, thus offering another path of length 11.

Crossrefs

Cf. A003415, A046099, A051674, A051903, A068346, A276086, A276087, A327859, A327860, A328099, A328112, A328114, A328116, A328307.
Cf. A328324 (a sequence giving upper bounds, computed with restricted search space).
Sequences for whose terms k, value a(k) has a guaranteed constant upper bound: A000040, A002110, A143293, A157037, A192192, A327978, A328232, A328233, A328239, A328240, A328243, A328249, A328313.
Sequences for whose terms k, it is guaranteed that a(k) has finite value > 0, even if not bound by a constant: A099308, A328116.
Cf. also A256750, A327966, A328110, A351029, A351088, A351067, A351071, A351073, A351089 and A351255, A351256, A351257, A351258, A351261.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327969(n,searchlim=0) = if(!n,n,my(xs=Set([n]),newxs,a,b,u); for(k=1,oo, print("n=", n, " k=", k, " xs=", xs); newxs=Set([]); for(i=1,#xs,u = xs[i]; a = A003415(u); if(0==a, return(k)); if(isprime(a), return(k+2)); b = A276086(u); if(isprime(b), return(k+1+(u>2))); newxs = setunion([a],newxs); if(!searchlim || (b<=searchlim),newxs = setunion([b],newxs))); xs = newxs));

Formula

a(0) = 0, a(p^p) = 1 + a(A276086(p^p)) for primes p, and for other numbers, a(n) = 1+min(a(A003415(n)), a(A276086(n))).
a(p) = 2 for all primes p.
For all n, a(n) <= A328324(n).
Let A stand the transition x -> A003415(x), and B stand for x -> A276086(x). The following sequences give some constant upper limits, because it is guaranteed that the combination given in brackets (the leftmost A or B is applied first) will always lead to a prime:
For all n, a(A157037(n)) = 3. [A]
For n > 1, a(A002110(n)) = 3. [B]
For all n, a(A192192(n)) <= 4. [AA]
For all n, a(A327978(n)) = 4. [AB]
For all n, a(A328233(n)) <= 4. [BA]
For all n, a(A143293(n)) <= 4. [BB]
For all n, a(A328239(n)) <= 5. [AAA]
For all n, a(A328240(n)) <= 5. [BAA]
For all n, a(A328243(n)) <= 5. [ABB]
For all n, a(A328313(n)) <= 5. [BBB]
For all n, a(A328249(n)) <= 6. [BAAA]
For all k in A046099, a(k) >= 4, and if A328114(k) > 1, then certainly a(k) > 4.

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

Views

Author

Reinhard Zumkeller, Feb 22 2009

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

  • Haskell
    a157037 n = a157037_list !! (n-1)
a157037_list = filter ((== 1) . a010051' . a003415) [1..]
-- Reinhard Zumkeller, Apr 08 2015
  • 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]]]]]; Select[Range[500], dn[dn[#]] == 1 &] (* T. D. Noe, Mar 07 2013 *)
  • PARI
    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
  • Python
    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

Formula

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

A327862 Numbers whose arithmetic derivative is of the form 4k+2, cf. A003415.

Original entry on oeis.org

9, 21, 25, 33, 49, 57, 65, 69, 77, 85, 93, 121, 129, 133, 135, 141, 145, 161, 169, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265, 289, 301, 305, 309, 315, 321, 329, 341, 351, 361, 365, 375, 377, 381, 393, 413, 417, 437, 445, 453, 459, 469, 473, 481, 485, 489, 493, 495, 497, 501, 505, 517, 529, 533, 537
Offset: 1

Views

Author

Antti Karttunen, Sep 30 2019

Keywords

Comments

All terms are odd because the terms A068719 are either multiples of 4 or odd numbers.
Odd numbers k for which A064989(k) is one of the terms of A358762. - Antti Karttunen, Nov 30 2022
The second arithmetic derivative (A068346) of these numbers is odd. See A235991. - Antti Karttunen, Feb 06 2024

Links

Crossrefs

Setwise difference A235992 \ A327864.
Setwise difference A046337 \ A360110.
Union of A369661 (k' has an even number of prime factors) and A369662 (k' has an odd number of prime factors).
Subsequences: A001248 (from its second term onward), A108181, A327978, A366890 (when sorted into ascending order), A368696, A368697.
Cf. A003415, A064989, A068346, A068719, A327863, A327865, A353495 (characteristic function).
Cf. also A235991, A358762, A358772, A358774.

Programs

  • PARI
    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
isA327862(n) = (2==(A003415(n)%4));
k=1; n=0; while(k<105, if(isA327862(n), print1(n, ", "); k++); n++);

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

Views

Author

Antti Karttunen, Oct 10 2019

Keywords

Comments

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).

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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)); };

A328233 Numbers n such that the arithmetic derivative of A276086(n) is prime.

Original entry on oeis.org

3, 7, 9, 33, 37, 38, 211, 213, 218, 241, 242, 246, 247, 249, 2313, 2317, 2319, 2341, 2342, 2346, 2521, 2523, 2526, 2529, 2550, 2553, 2559, 30031, 30038, 30039, 30061, 30062, 30063, 30066, 30069, 30242, 30243, 30249, 30270, 30278, 30279, 32341, 32342, 32347, 32370, 32373, 32377, 32379, 32551, 32553, 510513, 510518, 510519
Offset: 1

Views

Author

Antti Karttunen, Oct 09 2019

Keywords

Comments

Numbers n for which A327860(n) = A003415(A276086(n)) is a prime.
Numbers n such that A276086(n) is in A157037.
Terms come in distinct "batches", where in each batch they are "slightly more" than the nearest primorial (A002110) below. This is explained by the fact that for A276086(n) to be a squarefree (which is the necessary condition for A157037), n's primorial base expansion (A049345) must not contain digits larger than 1. Thus this is a subsequence of A276156.
Numbers n such that A327860(A276086(n)) = A003415(A276087(n)) is a prime [A276087(n) is in A157037] are much rarer: 2, 4, 30, 212, 421, 30045, 510511, 512820, 9729723, ...
For all terms k in this sequence, A327969(k) <= 4, and particularly A327969(k) = 2 when k is a prime. Otherwise, when k is not a prime, but A003415(k) is, A327969(k) = 3, while for other cases (when k is neither prime nor in A157037), we have A327969(k) = 4.

Links

Crossrefs

Cf. A002110, A003415, A051674, A157037, A276086, A327860, A327969, A327978, A328232, A328240.
Subsequence of A276156, of A328116, and of A328242.

Programs

  • PARI
    A327860(n) = { my(m=1, i=0, s=0, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), my(e=((n%nextpr)/pr)); m *= (prime(i)^e); s += (e / prime(i)); n-=(n%nextpr)); pr=nextpr); (s*m); };
isA328233(n) = isprime(A327860(n));

A328393 Numbers whose arithmetic derivative (A003415) is a squarefree number (A005117).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 10, 11, 13, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 37, 38, 41, 42, 43, 45, 47, 49, 53, 57, 58, 59, 61, 62, 63, 66, 67, 69, 70, 71, 73, 74, 75, 78, 79, 82, 83, 85, 89, 90, 93, 97, 98, 101, 102, 103, 105, 106, 107, 109, 110, 113, 114, 117, 118, 121, 126, 127, 129, 130, 131, 133, 134, 137, 139, 142
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2019

Keywords

Links

Crossrefs

Union of A000040 and A328234. Complement of A328303.
Cf. A005117, A008966, A003415.
Cf. A328252 (nonsquarefree terms), A157037, A192192, A327978 (other subsequences).

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328393(n) = issquarefree(A003415(n));

A351029 Number of integers whose arithmetic derivative is equal to the n-th primorial.

Original entry on oeis.org

0, 1, 3, 19, 114, 905, 9494, 124181, 2044847, 43755729, 1043468388, 30309948250
Offset: 1

Views

Author

Antti Karttunen, Feb 01 2022

Keywords

Comments

Number of integers k such that A003415(k) = A002110(n).
a(7) = A116979(7) + 1 since 1547371'=510510 and 1547371=7^2*23*1373 and every other example has only two prime factors. a(8) > A116979(8) because there is at least one term k in A327978 for which A003415(k) = 9699690 = A002110(8), which is not semiprime, that k being 79332523 = 17^2 * 277 * 991. - Edited by Craig J. Beisel, Sep 13 2022 and Antti Karttunen, Jan 05 2023
Most such k are semiprimes, i.e., are "Goldbachian solutions", counted by A116979. The non-semiprime solutions (A366890) form a very tiny minority, and are counted by A369000. - Antti Karttunen, Jan 19 2024

Examples

			a(1) = 0 because there are no such k that A003415(k) = 2 = A002110(1).
a(2) = 1 because there is only one number, 9, such that A003415(9) = A002110(2) = 6.
a(3) = 3 because there are exactly three numbers, k = 161, 209, 221, for which A003415(k) = A002110(3) = 30. (See A327978). These are all semiprime solutions, generated by the partitions of 30 into 2 primes: 30 = 7 + 23 = 11 + 19 = 13 + 17, and we have 7*23 = 161; 11*19 = 209; 13*17 = 221.

Links

Crossrefs

Cf. A002110, A002620, A003415, A099302, A099303, A116979, A327978, A366890 (nonsemiprime solutions), A368703 (the least of solutions), A368704 (the largest of solutions), A369000.
Cf. also A369239.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
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]));
A351029(n) = { my(g=A002110(n)); sum(k=1,A002620(g),A003415(k)==g); }; \\ Very naive and slow. See comments in A327978.
  • PARI
    A351029(n) = {v=prod(j=1,n,prime(j)); c=0; for(k=2, v^2/4, d=0; m=factor(k); for(i=1, matsize(m)[1], d+=(m[i,2]/m[i,1])*k; if(d>v, break;); ); if(d==v, c=c+1; ); ); c;} \\ Craig J. Beisel, Sep 13 2022

Formula

a(n) = Sum_{k=1..A002620(A002110(n))} [A003415(k) = A002110(n)], where [ ] is the Iverson bracket.
a(n) = A116979(n) + A369000(n). - Antti Karttunen, Jan 19 2024

Extensions

a(7) from Craig J. Beisel, Sep 13 2022
a(8)..a(12) [the last based on the value of A116979(12)] from Antti Karttunen, Jan 09 2024

A328243 Numbers whose arithmetic derivative (A003415) is larger than 1 and one of the terms of A143293 (partial sums of primorials).

Original entry on oeis.org

14, 45, 74, 198, 5114, 10295, 65174, 1086194, 20485574, 40354813, 465779078, 12101385979, 15237604243, 18046312939, 29501083259, 52467636437, 65794608773, 86725630997, 87741700037, 131833085077, 168380217557, 176203950283, 177332276971, 226152989747, 292546582253
Offset: 1

Views

Author

Antti Karttunen, Oct 10 2019

Keywords

Comments

From David A. Corneth, Oct 12 2019: (Start)
Let k' be the arithmetic derivative of k. Then to find terms of the form k = p * q where p, q are prime, we could see that k' = p + q. Then as one of them needs to be two, say p, needs to be 2, we have q = A143293(m) - 2 a prime. This would give terms 2 * q.
If terms are of the form k = p * q * r where p, q, r are distinct primes then k' = p*q + p*r + q*r. For m we like, we could solve p*q + p*r + q*r = A143293(m). checking p * q below some bound, we can solve for r and get r = (A143293(m) - p*q) / (p + q). With some extra constraints and searching different prime signatures, one might confirm terms found are all below some chosen upper bound. (End)
See sequences A369239 and A369240 for more observations and insights about the terms of this sequence. - Antti Karttunen, Jan 22 2024

Links

Crossrefs

Sequence A369240 sorted into ascending order.
Cf. A003415, A143293, A327969, A327978, A328313, A369239.

Programs

  • PARI
    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]));
A143293(n) = if(n==0, 1, my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); (s)); \\ From A143293.
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
isA328243(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(A276086(u)))); }; \\ This is very slow program!
k=0; for(n=1,A002620(A143293(6)),if(isA328243(n), k++; print1(n,", ")));

Formula

A327969(a(n)) <= 5 for all n.

Extensions

a(12)-a(25) from David A. Corneth and Giovanni Resta, Oct 12 2019
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