Displaying 1-10 of 19 results found.
Numbers m whose k-th arithmetic derivative is zero for some k. Complement of A099309.
+20
28
0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 29, 30, 31, 33, 34, 37, 38, 41, 42, 43, 46, 47, 49, 53, 57, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 77, 78, 79, 82, 83, 85, 89, 93, 94, 97, 98, 101, 103, 105, 107, 109, 113, 114, 118, 121, 126, 127, 129, 130
COMMENTS
The first derivative of 0 and 1 is 0. The second derivative of a prime number is 0.
EXAMPLE
18 is on this list because the first through fifth derivatives are 21, 10, 7, 1, 0.
MATHEMATICA
dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; nLim=200; lst={1}; i=1; While[i<=Length[lst], currN=lst[[i]]; pre=Intersection[Flatten[Position[d1, currN]], Range[nLim]]; pre=Complement[pre, lst]; lst=Join[lst, pre]; i++ ]; Union[lst]
PROG
(PARI)
\\ The following program would get stuck in nontrivial loops. However, we assume that the conjecture 3 in Ufnarovski & Åhlander paper holds ("The differential equation n^(k) = n has only trivial solutions p^p for primes p").
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));
isA099308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n)); \\ Antti Karttunen, Feb 14 2022
CROSSREFS
Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero), A099309 (complement, numbers whose k-th arithmetic derivative is nonzero for all k), A351078 (first noncomposite reached when iterating the derivative from these numbers), A351079 (the largest term on such paths). Cf. also A256750 (number of iterations needed to reach either 0 or a number with a factor of the form p^p), A327969, A351088.
a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(m*n) = m*a(n) + n*a(m).
(Formerly M3196)
+10
1065
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
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 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 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
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).
FORMULA
If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
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 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)
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)
EXAMPLE
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 + ...
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;
local a, f;
a := 0 ;
for f in ifactors(n)[2] do
a := a+ op(2, f)/op(1, f);
end do;
n*a ;
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 *)
PROG
(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) 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 (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
(Magma) Ad:=func<h | h*(&+[Factorisation(h)[i][2]/Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [n le 1 select 0 else Ad(n): n in [0..80]]; // Bruno Berselli, Oct 22 2013 (Python)
from sympy import factorint
return sum([int(n*e/p) for p, e in factorint(n).items()]) if n > 1 else 0
(Sage)
F = [] if n == 0 else factor(n)
return n * sum(g / f for f, g in F)
(GAP)
A003415:= Concatenation([0, 0], List(List([2..10^3], Factors),
(APL, Dyalog dialect) A003415 ← { ⍺←(0 1 2) ⋄ ⍵≤1:⊃⍺ ⋄ 0=(3⊃⍺)|⍵:((⊃⍺+(2⊃⍺)×(⍵÷3⊃⍺)) ((2⊃⍺)×(3⊃⍺)) (3⊃⍺)) ∇ ⍵÷3⊃⍺ ⋄ ((⊃⍺) (2⊃⍺) (1+(3⊃⍺))) ∇ ⍵} ⍝ Antti Karttunen, Feb 18 2024
CROSSREFS
Cf. A086134 (least prime factor of n'). Cf. A086131 (greatest prime factor of n'). Cf. A027471 (derivative of 3^(n-1)). Cf. A068237 (numerator of derivative of 1/n). Cf. A068238 (denominator of derivative of 1/n). Cf. A068328 (derivative of squarefree numbers). 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. 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. 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. 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. A229501 (n divides the n-th partial sum). 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. A305809 (Dirichlet convolution square). Cf. A069359 (similar formula which agrees on squarefree numbers). Cf. A258851 (the pi-based arithmetic derivative of n). Cf. A038554 (another sequence using "derivative" in its name, but involving binary expansion of n).
Start with n, and repeatedly apply the arithmetic derivative A003415. |a(n)| = the number of iterations to reach 0 (then a(n) is taken nonnegative) or a number having a factor of the form p^p with prime p, in which case a(n) = -|a(n)|.
+10
18
0, 1, 2, 2, 0, 2, 3, 2, 0, 4, 3, 2, 0, 2, 5, -1, 0, 2, 5, 2, 0, 4, 3, 2, 0, 4, -2, 0, 0, 2, 3, 2, 0, 6, 3, -1, 0, 2, 5, -1, 0, 2, 3, 2, 0, -2, 5, 2, 0, 6, -3, -1, 0, 2, 0, -1, 0, 4, 3, 2, 0, 2, 7, -2, 0, 6, 3, 2, 0, -3, 3, 2, 0, 2, -2, -2, 0, 6, 3, 2, 0, 0, 3, 2, 0, 4, -3, -1, 0, 2, -2, -1, 0, 4, 7, -1, 0, 2, 7, -3
COMMENTS
Under iterations of the arithmetic derivative, the orbit of some numbers ends in zero, and the orbit of all others (I conjecture) reaches a number of the form m*p^p with prime p, from where on it keeps this form and grows to infinity iff m>1, or remains at this fixed point if m=1.
This is an extension of the sequence A099307 which counts the steps to reach 0 or yields 0 if this never happens.
FORMULA
a(n) = 0 <=> n = 0 or n = m*p^p for some prime p and some m >= 1 (which is a fixed point iff m = 1).
a(n) = 1 <=> n = 1.
a(n) = 2 <=> n is prime.
a(n) <= 0 <=> n is in A099309 U {0}. If n > 0, the iterations of A003415 applied to n end in a nonzero fixed point or grow to infinity. a(n) > 0 <=> n is in A099308 \ {0}.
MATHEMATICA
w = {}; nn = 2^16; k = 1; While[Set[m, #^#] <= nn &[Prime[k]], AppendTo[w, m]; k++]; a3415[n_] := a3415[n] = Which[n < 2, 0, PrimeQ[n], 1, True, n Total[#2/#1 & @@@ FactorInteger[n]]]{0, 1}~Join~Reap[Do[Which[PrimeQ[n], Sow[2], MemberQ[w, n], Sow[0], True, Sow@ If[#[[-1]] == 0, Length[#] - 1, -Length[#] + 1] &[NestWhileList[a3415, n, And[! Divisible[#, 4], FreeQ[w, #]] &, 1]]], {n, 2, nn}] ][[-1, -1]] (* Michael De Vlieger, Jan 04 2023 *)
PROG
(PARI) a(n, c=0)={n&&until(!n=factorback(n~)*sum(i=1, #n, n[2, i]/n[1, i]), for(i=1, #n=factor(n)~, n[1, i]>n[2, i]||return(-c)); c++); c}
CROSSREFS
Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero), A099308 (numbers whose k-th arithmetic derivative is zero for some k, positions of terms > 0 after the initial 0), A099309 (numbers whose k-th arithmetic derivative is nonzero for all k, positions of terms <= 0 after the initial 0), A359547 (positions of negative terms), A327934 (positions of -1's).
Numbers for which there is at least one such prime p that p^p divides the arithmetic derivative of n, A003415(n).
+10
18
4, 8, 12, 15, 16, 20, 24, 27, 28, 32, 35, 36, 39, 40, 44, 48, 51, 52, 54, 55, 56, 60, 64, 68, 72, 76, 80, 81, 84, 87, 88, 91, 92, 95, 96, 100, 104, 108, 111, 112, 115, 116, 119, 120, 123, 124, 128, 132, 135, 136, 140, 143, 144, 148, 152, 155, 156, 158, 159, 160, 162, 164, 168, 172, 176, 180, 183, 184, 187, 188, 189, 192, 196, 200
COMMENTS
For all n, A003415(a(n)) is also present in this sequence and A003415(a(n)) = m * A327965(a(n)) for some m > 1.
PROG
(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
A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
CROSSREFS
Indices of nonzero terms in A327928.
Least k such that the k-th arithmetic derivative of n is zero, or 0 if no k exists.
+10
16
1, 2, 2, 0, 2, 3, 2, 0, 4, 3, 2, 0, 2, 5, 0, 0, 2, 5, 2, 0, 4, 3, 2, 0, 4, 0, 0, 0, 2, 3, 2, 0, 6, 3, 0, 0, 2, 5, 0, 0, 2, 3, 2, 0, 0, 5, 2, 0, 6, 0, 0, 0, 2, 0, 0, 0, 4, 3, 2, 0, 2, 7, 0, 0, 6, 3, 2, 0, 0, 3, 2, 0, 2, 0, 0, 0, 6, 3, 2, 0, 0, 3, 2, 0, 4, 0, 0, 0, 2, 0, 0, 0, 4, 7, 0, 0, 2, 7, 0, 0, 2, 0, 2, 0, 3
COMMENTS
Denote the k-th derivative of n by d(n,k). We know that we can stop taking derivatives if either d(n,k) = 0 or d(n,k) has a factor of the form p^p for prime p. In the latter case, the derivatives will stay constant or grow without bound.
MATHEMATICA
dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[k=0; d=n; done=False; While[If[d==1, done=True, f=FactorInteger[d]; Do[If[f[[i, 1]]<=f[[i, 2]], done=True], {i, Length[f]}]]; !done, k++; d=dn[d]]; If[d==1, k+1, 0], {n, 200}]
CROSSREFS
Cf. A003415 (arithmetic derivative of n). Cf. A099308 (numbers whose k-th arithmetic derivative is zero for some k). Cf. A099309 (numbers whose k-th arithmetic derivative is nonzero for all k). Cf. A189760 (least number whose n-th arithmetic derivative is zero).
a(n) = 1 if k-th arithmetic derivative of n is zero for some k, otherwise 0.
+10
16
1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1
LINKS
Michael De Vlieger, Bitmap of a(n), n = 0..2^24, 2048 X 2048 pixels, with 0 in white and 1 in black. Furnishes 4260302 terms of A099308.
FORMULA
For prime p, a(p) = 1, a(p^p * m) = 0, for all m >= 1. a(4m) = 0 for m > 0. - Michael De Vlieger, Jan 04 2023 a(0) = 1; and for n > 0, a(n) = A359550(n) * a( A003415(n)). [Provided that Conjecture 3 given on page 5 of Ufnarovski and Åhlander 2003 paper holds, i.e., that map x -> x' never forms nontrivial loops] (End)
MATHEMATICA
w = {}; nn = 2^10; k = 1; While[Set[m, #^#] <= nn &[Prime[k]], AppendTo[w, m]; k++]; a3415[n_] := a3415[n] = Which[Abs@ n < 2, 0, PrimeQ[n], 1, True, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]{1, 1}~Join~Reap[Do[Which[PrimeQ[n], Sow[1], MemberQ[w, n], Sow[0], True, If[NestWhileList[a3415, n, And[! Divisible[#, 4], FreeQ[w, #]] &, 1][[-1]] == 0, Sow[1], Sow[0]]], {n, 2, nn}]][[-1, -1]] (* Michael De Vlieger, Jan 04 2023 *) (* 2nd program: generate m <= 2^24 terms of the sequence from the bitmap above: *)
m = 10^3; Flatten[ImageData[Import["https://oeis.org/ A328308/a328308.png"], "Bit"]][[1 ;; m]] /. {0 -> 1, 1 -> 0} (* Michael De Vlieger, Jan 04 2023 *)
PROG
(PARI)
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));
A328308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n));
CROSSREFS
Characteristic function of A099308. Cf. A003415, A099309 (positions of zeros), A256750, A328306 [= a( A276086(n))], A328309 (partial sums), A341996, A341999 (one's complement), A342023, A351071, A359541 (inverse Möbius transform), A359543, A359546, A359550, A368915.
a(n) = 1 if the arithmetic derivative of n is a multiple of 4, otherwise 0.
+10
14
1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0
FORMULA
a(n) = 1 if A353493(n) = 0, otherwise a(n) = 0.
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
CROSSREFS
Characteristic function of A327864.
a(n) = 1 if the k-th arithmetic derivative is nonzero for all k >= 0, otherwise 0.
+10
13
0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0
COMMENTS
Characteristic function of A099309.
FORMULA
a(n) = 1 if n is in A100716 or ends there by repeated applications of A003415, otherwise a(n) = 0 (when n instead reaches 0 by such iteration). For all n > 0, a(n) = [ A256750(n) < 1]. For all n > 0, a(n) >= [ A129251(n)>0], i.e., if A129251(n) is nonzero, then certainly a(n) = 1. For all n > 1, a(n) >= [ A341997(n) > 1].
PROG
(PARI)
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));
A341999(n) = if(!n, n, while(n>1, n = A003415checked(n)); (!n));
The n-th arithmetic derivative of 2^3.
+10
12
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
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
FORMULA
a(n+1) = A003415(a(n)), a(0) = 2^3 = 8.
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 *)
PROG
(Haskell)
a129150 n = a129150_list !! n
(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
Numbers k such that there is no prime p such that p^p divides k, but for its arithmetic derivative a positive finite number of such primes exist.
+10
11
15, 35, 39, 51, 55, 87, 91, 95, 111, 115, 119, 123, 143, 155, 158, 159, 183, 187, 203, 215, 219, 225, 235, 247, 259, 267, 275, 287, 291, 295, 299, 303, 319, 323, 327, 329, 335, 339, 355, 371, 374, 391, 395, 403, 407, 410, 411, 415, 427, 441, 447, 451, 471, 473, 482, 511, 515, 519, 525, 527, 533, 535, 543, 551, 559, 579
COMMENTS
a(15) = 158, being even, is the first term not in A080774 or A202237, while 275 (= 5^2 * 11) is the first odd term not in the latter. After 1, 3375 (= 3^3 * 5^3) is the first term in A202237 that is not present in this sequence, even though the overwhelming majority of the terms of A202237 are also included here. On the other hand, A080774 is a genuine subsequence of this sequence, as the sum of prime factors of such semiprimes is always a multiple of 4. In contrast to A360110, this is not a multiplicative semigroup; For example, although 15 = 3*5 and 275 = 5^2 * 11 are both present, their product 15*275 = 4125 = 3 * 5^3 * 11 is not. - Antti Karttunen, Jan 31 2023
EXAMPLE
1 has arithmetic derivative 1' = A003415(1) = 0. There are an infinite number of primes p such that p^p | 0, but because infinity is not a finite number, 1 is excluded from this sequence. 275 = 5^2 * 11 has no divisor of the form p^p, but its arithmetic derivative 275' = 135 = 3^3 * 5 has one divisor of the form p^p, therefore 275 is included in this sequence.
4125 = 3 * 5^3 * 11 has arithmetic derivative 4125' = A003415(4125) = 4225 = 5^2 * 13^2, that has no divisor of the form p^p, therefore 4125 is not included in this sequence. (End)
PROG
(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
A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
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