A369008 -id:A369008 - cp's OEIS frontend

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

Offset: 1

Reinhard Zumkeller, Apr 25 2003

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)

			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.

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

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.

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

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

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

Secondary definition added by Antti Karttunen, Nov 12 2024

A381588 If n = Product (p_j^k_j) then a(n) = Product (lcm(p_j, k_j)), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 6, 10, 11, 6, 13, 14, 15, 4, 17, 12, 19, 10, 21, 22, 23, 18, 10, 26, 3, 14, 29, 30, 31, 10, 33, 34, 35, 12, 37, 38, 39, 30, 41, 42, 43, 22, 30, 46, 47, 12, 14, 20, 51, 26, 53, 6, 55, 42, 57, 58, 59, 30, 61, 62, 42, 6, 65, 66, 67, 34, 69, 70, 71

Offset: 1

Paolo Xausa, Feb 28 2025

			a(18) = 12 because 18 = 2^1*3^2, lcm(2,1) = 2, lcm(3,2) = 6 and 2*6 = 12.
a(300) = 30 because 300 = 2^2*3^1*5^2, lcm(2,2) = 2, lcm(3,1) = 3, lcm(5,2) = 10 and 2*3*10 = 60.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008473, A008477, A035306, A144338 (fixed points), A369008 (analogous for gcd).

A381588[n_] := Times @@ LCM @@@ FactorInteger[n];
Array[A381588, 100]

a(n) = my(f=factor(n)); prod(i=1, #f~, lcm(f[i,1], f[i,2])); \\ Michel Marcus, Mar 02 2025

a(p) = p, for p prime.

A369009 a(n) = A342001(n) - A083345(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 2, 8, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 7, 0, 0, 0, 14, 0, 6, 0, 0, 0, 0, 0, 23, 0, 0, 0, 3, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 11, 0, 0, 0, 31, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 10

Offset: 1

Antti Karttunen, Jan 14 2024

nonn

Arithmetic derivative of n without its "inherited divisor" minus the arithmetic derivative of n without its greatest common divisor with n.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A003415, A003557, A083345, A085731, A342001, A369008.

Cf. A342090 (positions of terms > 0).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A369009(n) = { my(u=A003415(n)); (u/A003557(n))-(u/gcd(n,u)); };

A381613 If n = Product (p_j^k_j) then a(n) = Product (min(p_j, k_j)), with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1

Offset: 1

Paolo Xausa, Mar 01 2025

First differs from A323308 at n = 27.

			a(18) = 2 because 18 = 2^1*3^2, min(2,1) = 1, min(3,2) = 2 and 1*2 = 2.
a(300) = 4 because 300 = 2^2*3^1*5^2, min(2,2) = 2, min(3,1) = 1, min(5,2) = 2 and 2*1*2 = 4.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008473, A008477, A035306, A323308, A369008, A381588, A381614.

A381613[n_] := Times @@ Min @@@ FactorInteger[n];
Array[A381613, 100]

a(n) = my(f=factor(n)); prod(i=1, #f~, min(f[i,1], f[i,2])); \\ Michel Marcus, Mar 02 2025

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (1/p - 1/p^p)/(p-1)) = 1.59383299054679951264... . - Amiram Eldar, Mar 07 2025

A381614 If n = Product (p_j^k_j) then a(n) = Product (max(p_j, k_j)), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 3, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19, 10, 21, 22, 23, 9, 5, 26, 3, 14, 29, 30, 31, 5, 33, 34, 35, 6, 37, 38, 39, 15, 41, 42, 43, 22, 15, 46, 47, 12, 7, 10, 51, 26, 53, 6, 55, 21, 57, 58, 59, 30, 61, 62, 21, 6, 65, 66, 67, 34, 69, 70, 71, 9, 73

Offset: 1

Paolo Xausa, Mar 01 2025

			a(18) = 6 because 18 = 2^1*3^2, max(2,1) = 2, max(3,2) = 3 and 2*3 = 6.
a(300) = 30 because 300 = 2^2*3^1*5^2, max(2,2) = 2, max(3,1) = 3, max(5,2) = 5 and 2*3*5 = 30.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008473, A008477, A035306, A065463, A369008, A381588, A381613.

A381614[n_] := Times @@ Max @@@ FactorInteger[n];
Array[A381614, 100]

a(n) = my(f=factor(n)); prod(i=1, #f~, max(f[i,1], f[i,2])); \\ Michel Marcus, Mar 02 2025

a(p) = p, for p prime.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065463 * Product_{p prime} (1 + 1/((p^2-1)*(p^2+p-1)*p^(2*p-2))) = 0.71628338157754073004... . - Amiram Eldar, Mar 07 2025

cp's OEIS Frontend

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.

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A381588 If n = Product (p_j^k_j) then a(n) = Product (lcm(p_j, k_j)), with a(1) = 1.

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A369009 a(n) = A342001(n) - A083345(n).

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A381613 If n = Product (p_j^k_j) then a(n) = Product (min(p_j, k_j)), with a(1) = 1.

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A381614 If n = Product (p_j^k_j) then a(n) = Product (max(p_j, k_j)), with a(1) = 1.

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