A083345 - OEIS

%I #68 Nov 12 2024 17:24:29

%S 0,1,1,1,1,5,1,3,2,7,1,4,1,9,8,2,1,7,1,6,10,13,1,11,2,15,1,8,1,31,1,5,

%T 14,19,12,5,1,21,16,17,1,41,1,12,13,25,1,7,2,9,20,14,1,3,16,23,22,31,

%U 1,23,1,33,17,3,18,61,1,18,26,59,1,13,1,39,11,20,18,71,1,11,4,43,1,31,22

%N Numerator of r(n) = Sum(e/p: n=Product(p^e)); a(n) = n' / gcd(n,n'), where n' is the arithmetic derivative of n.

%C Least common multiple of n and its arithmetic derivative, divided by n, i.e. a(n) = lcm(n,n')/n = A086130(n)/A000027(n). - _Giorgio Balzarotti_, Apr 14 2011

%C From _Antti Karttunen_, Nov 12 2024: (Start)

%C Positions of multiples of any natural number in this sequence (like A369002, A369644, A369005, or A369007) form always a multiplicative semigroup: if m and n are in that sequence, then so is m*n.

%C Proof: a(x) = x' / gcd(x,x') = A003415(x) / A085731(x) by definition. Let v_p(x) be the p-adic valuation of x, with p prime. Let e = v_p(c), the p-adic valuation of natural number c whose multiples we are searching for. For v_p(a(x)) >= e > 0 and v_p(a(y)) >= e > 0 to hold we must have v_p(x') = v_p(x)+h and v_p(y') = v_p(y)+k, for some h >= e, k >= e for p^e to divide a(x) and a(y).

%C   Then, as a(xy) = (xy)' / gcd(xy,(xy)') = (x'y + y'x) / gcd(xy, (x'y + y'x)), we have, for the top side, v_p((xy)') = min(v_p(x')+v_p(y), v_p(y')+v_p(x)) = min(v_p(x) + h + v_p(y), v_p(y) + k + v_p(x)) = v_p(xy) + min(h,k), and for the bottom side we get v_p(gcd(xy, (x'y + y'x))) = min(v_p(xy), v_p(xy) + min(h,k)) = v_p(xy), so v_p(a(xy)) = min(h,k) >= e, thus p^e | a(xy). For a composite c that is not a prime power, c | a(xy) holds if the above equations hold for all p^e || c.

%C (End)

%H Antti Karttunen, <a href="/A083345/b083345.txt">Table of n, a(n) for n = 1..65537</a>

%F The fraction a(n)/A083346(n) is totally additive with a(p) = 1/p. - _Franklin T. Adams-Watters_, May 17 2006

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A083346(k) = Sum_{p prime} 1/(p*(p-1)) = 0.773156... (A136141). - _Amiram Eldar_, Sep 29 2023

%F a(n) = A003415(n) / A085731(n) = A342001(n) / A369008(n). - _Antti Karttunen_, Jan 16 2024

%e Fractions begin with 0, 1/2, 1/3, 1, 1/5, 5/6, 1/7, 3/2, 2/3, 7/10, 1/11, 4/3, ...

%e For n = 12, 2*2*3 = 2^2 * 3^1 --> r(12) = 2/2 + 1/3 = (6+2)/6, therefore a(12) = 4, A083346(12) = 3.

%e For n = 18, 2*3*3 = 2^1 * 3^2 --> r(18) = 1/2 + 2/3 = (3+4)/6, therefore a(18) = 7, A083346(18) = 6.

%t Array[Numerator@ Total[FactorInteger[#] /. {p_, e_} /; e > 0 :> e/p] - Boole[# == 1] &, 85] (* _Michael De Vlieger_, Feb 25 2018 *)

%o (PARI) A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~,i,f[i,2]/f[i,1]))); }; \\ _Antti Karttunen_, Feb 25 2018

%Y Cf. A083346 (denominator), A000027, A072873, A083347, A083348, A085731, A086130, A136141, A342001, A342002 [= a(A276086(n))].

%Y Cf. A369001 (anti-parity), A377874 (parity).

%Y Cf. A369002 (positions of even terms), A369003 (of odd terms), A369644 (of multiples of 3), A369005 (of multiples of 4), A373265 (of terms of the form 4m+2), A369007 (of multiples of 27), A369008, A369068 (Möbius transform), A369069.

%Y Cf. A373363, A373366, A373377, A373485.

%K nonn,easy,frac

%O 1,6

%A _Reinhard Zumkeller_, Apr 25 2003

%E Secondary definition added by _Antti Karttunen_, Nov 12 2024