A083345 - OEIS

A083345

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.

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, 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, 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

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OFFSET

1,6

COMMENTS

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

From Antti Karttunen, Nov 12 2024: (Start)

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.

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).

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.

(End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

The fraction a(n)/A083346(n) is totally additive with a(p) = 1/p. - Franklin T. Adams-Watters, May 17 2006

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

a(n) = A003415(n) / A085731(n) = A342001(n) / A369008(n). - Antti Karttunen, Jan 16 2024

EXAMPLE

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, ...

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.

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.

MATHEMATICA

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

PROG

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

CROSSREFS

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

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

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.

Cf. A373363, A373366, A373377, A373485.

Sequence in context: A136744 A068237 A373363 * A087262 A082343 A166125

Adjacent sequences: A083342 A083343 A083344 * A083346 A083347 A083348

KEYWORD

nonn,easy,frac

AUTHOR

Reinhard Zumkeller, Apr 25 2003

EXTENSIONS

Secondary definition added by Antti Karttunen, Nov 12 2024

STATUS

approved