A369069 -id:A369069 - 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

A369067 Lexicographically earliest infinite sequence such that a(i) = a(j) => A369066(i) = A369066(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 2, 3, 2, 4, 5, 1, 2, 6, 1, 7, 2, 8, 9, 1, 10, 11, 4, 9, 2, 6, 3, 1, 6, 1, 12, 5, 2, 13, 14, 1, 11, 15, 8, 11, 16, 17, 18, 1, 8, 19, 1, 20, 2, 6, 3, 1, 4, 4, 6, 4, 6, 5, 9, 1, 6, 8, 2, 20, 2, 21, 11, 1, 16, 22, 13, 15, 23, 24, 25, 1, 13, 21, 1, 26, 27, 28, 29, 1, 30, 29, 17, 31, 13, 21, 15, 1, 16

Offset: 0

Antti Karttunen, Jan 16 2024

nonn

Restricted growth sequence transform of A369066.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A005940, A008836, A083345, A369066, A369069.

Cf. also A366805 (compare the scatter plots).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A008836(n) = ((-1)^bigomega(n));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369069(n) = sumdiv(n,d,A008836(n/d)*A083345(d));
A369066(n) = A369069(A005940(1+n));
v369067 = rgs_transform(vector(1+up_to,n,A369066(n-1)));
A369067(n) = v369067[1+n];

A369068 Möbius transform of A083345, where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 1, 2, 1, 5, 1, -1, 1, 7, 6, -1, 1, 1, 1, -1, 8, 11, 1, 5, 1, 13, -1, -1, 1, 14, 1, 3, 12, 17, 10, -1, 1, 19, 14, 9, 1, 20, 1, -1, 4, 23, 1, -3, 1, 1, 18, -1, 1, -3, 14, 13, 20, 29, 1, -6, 1, 31, 6, -2, 16, 32, 1, -1, 24, 34, 1, 1, 1, 37, 2, -1, 16, 38, 1, -5, 3, 41, 1, -8, 20, 43, 30, 21, 1, 4, 18

Offset: 1

Antti Karttunen, Jan 16 2024

sign

In contrast to A346485, this sequence has also negative values. Compare also to A369069.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003415, A008683, A083345.

Cf. also A346485, A369069.

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369068(n) = sumdiv(n,d,moebius(n/d)*A083345(d));

a(n) = Sum_{d|n} A008683(n/d) * A083345(d).

A369066 Dirichlet convolution of Liouville's lambda (A008836) with A083345, as reordered by the Doudna sequence.

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 1, 3, 1, 5, 6, 0, 1, 2, 0, -1, 1, 7, 8, 0, 10, 14, 5, 8, 1, 2, 3, 0, 2, 0, 4, 6, 1, 11, 12, 0, 14, 20, 7, 14, 16, 34, 44, 0, 7, 9, 0, -3, 1, 2, 3, 0, 5, 5, 2, 5, 2, 6, 8, 0, 2, 7, 1, -3, 1, 13, 14, 0, 16, 32, 11, 20, 18, 54, 68, 0, 11, 13, 0, -5, 22, 76, 92, 0, 124, 92, 34, 36, 11, 13, 20, 0, 16

Offset: 0

Antti Karttunen, Jan 16 2024

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A005940, A008836, A083345, A369067 (rgs-transform), A369069.

Cf. also A366803 (compare the scatter plots).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A008836(n) = ((-1)^bigomega(n));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369069(n) = sumdiv(n,d,A008836(n/d)*A083345(d));
A369066(n) = A369069(A005940(1+n));

a(n) = A369069(A005940(1+n)).

A369455 Dirichlet convolution of A083345 with A055615 (Dirichlet inverse of n), where A083345(n) = (n'/gcd(n,n')) and n' is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, -1, 1, 0, 1, 1, -1, 0, 1, -3, 1, 0, 0, -4, 1, -6, 1, -3, 0, 0, 1, 0, -3, 0, -5, -3, 1, 0, 1, 1, 0, 0, 0, 9, 1, 0, 0, 0, 1, 0, 1, -3, -6, 0, 1, -3, -5, -20, 0, -3, 1, -8, 0, 0, 0, 0, 1, 0, 1, 0, -6, -7, 0, 0, 1, -3, 0, 0, 1, -6, 1, 0, -20, -3, 0, 0, 1, -3, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -3, 0, 0, 0, 0, 1, -42, -6, 29

Offset: 1

Antti Karttunen, Jan 27 2024

sign

Dirichlet convolution of this sequence with A000010 (Euler phi) is A369068.

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

Cf. A055615, A083345.

Cf. also A349394, A349396, A369068, A369069.

A055615(n) = (n*moebius(n));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369455(n) = sumdiv(n, d, A083345(n/d)*A055615(d));

a(n) = Sum_{d|n} A055615(n/d)*A083345(d).

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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A369067 Lexicographically earliest infinite sequence such that a(i) = a(j) => A369066(i) = A369066(j) for all i, j >= 0.

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A369068 Möbius transform of A083345, where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.

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A369066 Dirichlet convolution of Liouville's lambda (A008836) with A083345, as reordered by the Doudna sequence.

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A369455 Dirichlet convolution of A083345 with A055615 (Dirichlet inverse of n), where A083345(n) = (n'/gcd(n,n')) and n' is the arithmetic derivative of n.

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