A083346 -id:A083346 - cp's OEIS frontend

A359588 Dirichlet inverse of A083346.

1, -2, -3, 3, -5, 6, -7, -6, 6, 10, -11, -9, -13, 14, 15, 12, -17, -12, -19, -15, 21, 22, -23, 18, 20, 26, -10, -21, -29, -30, -31, -24, 33, 34, 35, 18, -37, 38, 39, 30, -41, -42, -43, -33, -30, 46, -47, -36, 42, -40, 51, -39, -53, 20, 55, 42, 57, 58, -59, 45, -61, 62, -42, 48, 65, -66, -67, -51, 69

Offset: 1

Antti Karttunen, Jan 09 2023

Multiplicative because A083346 is.

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

Cf. A083346, A091428 (positions of odd terms), A359592 (parity of terms).

Cf. also A359577.

f[p_, e_] := If[Divisible[e, p], 1, p]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, May 18 2023 *)

A083346(n) = { my(f=factor(n)); denominator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
memoA359588 = Map();
A359588(n) = if(1==n,1,my(v); if(mapisdefined(memoA359588,n,&v), v, v = -sumdiv(n,d,if(dA083346(n/d)*A359588(d),0)); mapput(memoA359588,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083346(n/d) * a(d).

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.

Original entry on oeis.org

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

A085731 Greatest common divisor of n and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 3, 1, 1, 4, 1, 1, 1, 16, 1, 3, 1, 4, 1, 1, 1, 4, 5, 1, 27, 4, 1, 1, 1, 16, 1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 4, 3, 1, 1, 16, 7, 5, 1, 4, 1, 27, 1, 4, 1, 1, 1, 4, 1, 1, 3, 64, 1, 1, 1, 4, 1, 1, 1, 12, 1, 1, 5, 4, 1, 1, 1, 16, 27, 1, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 1, 1, 16

Offset: 1

Reinhard Zumkeller, Jul 20 2003

a(n) = 1 iff n is squarefree (A005117), cf. A068328.

This sequence is very probably multiplicative. - Mitch Harris, Apr 19 2005

T. D. Noe, Table of n, a(n) for n = 1..10000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Cf. A003415, A005117, A048103, A068328, A083346, A083347, A086130, A129251, A189100, A189036, A189103, A190116, A276086, A327858, A327938, A328572, A340070.

Cf. A072873 (positions of records), A268398 (partial sums).

a085731 n = gcd n $ a003415 n -- Reinhard Zumkeller, May 10 2011

d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; a[n_] := GCD[n, d[n]]; Table[a[n], {n, 1, 96}] (* Jean-François Alcover, Feb 21 2014 *)
f[p_, e_] := p^If[Divisible[e, p], e, e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 31 2023 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if (f[i,2] % f[i,1], f[i,2]--);); factorback(f);} \\ Michel Marcus, Feb 14 2016

a(n) = GCD(n, A003415(n)).
Multiplicative with a(p^e) = p^e if p divides e; a(p^e) = p^(e-1) otherwise. - Eric M. Schmidt, Oct 22 2013
From Antti Karttunen, Feb 28 2021: (Start)
Thus a(A276086(n)) = A328572(n), by the above formula and the fact that A276086 is a permutation of A048103.
a(n) = n / A083346(n) = A190116(n) / A086130(n). (End)

A328571 Primorial base expansion of n converted into its prime product form, but with all nonzero digits replaced by 1's: a(n) = A007947(A276086(n)).

Original entry on oeis.org

1, 2, 3, 6, 3, 6, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70

Offset: 0

Antti Karttunen, Oct 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A001221, A007947, A083346, A267263, A276086, A328572.

Cf. A276156 (gives the indices where this coincides with A276086).

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
a[n_] := rad[A276086[n]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen in A276086 *)

A328571(n) = { my(m=1, p=2); while(n, m *= (p^!!(n%p)); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A007947(A276086(n)).
a(n) = A276086(n) / A328572(n).
a(A276156(n)) = A276086(A276156(n)). [And at no other points the equality holds]
A001221(a(n)) = A267263(n).
a(n) = A083346(A276086(n)). - Antti Karttunen, Feb 28 2021

A083347 Numbers k such that Sum(e/p: k=Product(p^e)) < 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 1

Reinhard Zumkeller, Apr 25 2003

nonn

Numbers k whose arithmetic derivative (A003415) k' < k. - T. D. Noe, Apr 24 2011

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A003415, A072873, A051674, A083348.

Cf. A168036.

a083347 n = a083347_list !! (n-1)
a083347_list = filter ((< 0) . a168036) [1..]
-- Reinhard Zumkeller, May 22 2015, May 10 2011

Select[Range@ 102, If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger@ Abs@ #]] < # &] (* Michael De Vlieger, Feb 02 2019 *)

A083345(a(n)) < A083346(a(n));
A083345(A051674(n)) = A083346(A051674(n));
A083345(A083348(n)) > A083346(A083348(n)).
A168036(a(n)) < 0. - Reinhard Zumkeller, May 22 2015

A083348 Numbers k such that r(k) = Sum(e/p: k = Product(p^e)) > 1.

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 198

Offset: 1

Reinhard Zumkeller, Apr 25 2003

nonn

The number of terms not exceeding 10^m, for m = 1, 2, ..., are 1, 29, 318, 3174, 31763, 317813, 3177179, 31774009, 317745099, 3177373819, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3177... . - Amiram Eldar, Jun 24 2022

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A003415, A072873, A051674, A083345, A083346, A083347, A168036, A369048 (characteristic function), A369049.

Subsequence of A100717.

a083348 n = a083348_list !! (n-1)
a083348_list = filter ((> 0) . a168036) [1..]
-- Reinhard Zumkeller, May 22 2015, May 10 2011

ad[n_] := Switch[n, 0 | 1, 0, _, If[PrimeQ[n], 1, Sum[Module[ {p, e}, {p, e} = pe; n e/p], {pe, FactorInteger[n]}]]];
Select[Range[1000], ad[#] > # &] (* Jean-François Alcover, May 04 2023 *)

A083345(a(n)) > A083346(a(n)).
A083345(A051674(n)) = A083346(A051674(n)).
A083345(A083347(n)) < A083346(A083347(n)).
A168036(a(n)) > 0. - Reinhard Zumkeller, May 22 2015

A189100 a(n) = lcm(n!,n!')/gcd(n!,n!'), where n!' is the arithmetic derivative of n! (A068311).

Original entry on oeis.org

0, 2, 30, 66, 1830, 645, 33180, 198870, 228270, 64785, 7960260, 9738960, 1663226565, 7232635410, 857066210, 1057466410, 307311194190, 767464487790, 278292627277665, 306517823106495, 35302033071305, 147385363695570, 78207294248313230, 198777858520921680

Offset: 1

Giorgio Balzarotti, Apr 16 2011

nonn

Least common multiple of n! and its arithmetic derivative divided by greatest common divisor of n! and its arithmetic derivative.

			n = 5: 5! = 120, 120' = 244, gcd(120,244) = 4, lcm(120,244) = 7320, 7320/4 = 1830 -> a(5) = 1830.

Nathaniel Johnston, Table of n, a(n) for n = 1..200

A068311, A083346, A083346, A085731, A086130

a(n) = lcm(n!,n!')/gcd(n!,n!') = A086130(n!)/A085731(n!) = lcm(A000142(n),A068311(n))/gcd(A000142(n),A068311(n))

A189102 Greatest common divisor of n! and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 4, 4, 48, 48, 192, 1728, 34560, 34560, 414720, 414720, 2903040, 130636800, 2090188800, 2090188800, 25082265600, 25082265600, 501645312000, 31603654656000, 347640201216000, 347640201216000, 5562243219456000, 139056080486400000

Offset: 1

Giorgio Balzarotti, Apr 16 2011

nonn

			n = 5: 5! = 120, 120' = 244, gcd(120,244) = 4 -> a(5) = 4

Nathaniel Johnston, Table of n, a(n) for n = 1..200

A068311, A083346, A083346, A085731, A086130, A189100

a(n) = gcd(n!,n!') = gcd(A000142(n),A068311(n)).

A189036 a(n)= lcm(n,n')/gcd(n,n'), where n' is the arithmetic derivative of n.

Original entry on oeis.org

0, 2, 3, 1, 5, 30, 7, 6, 6, 70, 11, 12, 13, 126, 120, 2, 17, 42, 19, 30, 210, 286, 23, 66, 10, 390, 1, 56, 29, 930, 31, 10, 462, 646, 420, 15, 37, 798, 624, 170, 41, 1722, 43, 132, 195, 1150, 47, 21, 14, 90, 1020, 182, 53, 6, 880, 322, 1254, 1798, 59, 345, 61, 2046, 357, 3, 1170, 4026, 67, 306, 1794, 4130, 71, 78, 73, 2886, 165, 380, 1386, 5538, 79, 55, 12, 3526, 83, 651, 1870, 3870, 2784, 770, 89, 1230, 1820, 552, 3162, 4606, 2280, 102, 97, 154, 825, 35

Offset: 1

Giorgio Balzarotti, Apr 15 2011

nonn

Least common multiple of n and its arithmetic derivative divided by greatest common divisor of n and its arithmetic derivative.

			n = 8, n'= 12,  lcm(8,12)= 24, gcd(8,12)= 4, hence a(8)=24/4 = 6.

Cf. A003415, A085731, A086130, A083345, A083346.

a(n) = lcm(n,n')/gcd(n,n') = A086130(n)/A085731(n).

A189103 Least common multiple of n! and its arithmetic derivative.

Original entry on oeis.org

0, 2, 30, 264, 7320, 30960, 1592640, 38183040, 394450560, 2238969600, 275106585600, 4038941491200, 689773321036800, 20996629900646400, 111964387062528000, 2210304446558208000, 642338416210563072000, 19249748121316737024000, 6980209591900198477824000

Offset: 1

Giorgio Balzarotti, Apr 16 2011

nonn

			n = 5: 5! = 120, 120' = 244, lcm(120,244) = 7320 -> a(5) = 7320

Nathaniel Johnston, Table of n, a(n) for n = 1..200

Cf. A068311, A083346, A083346, A085731, A086130, A189100, A189102.

a(n) = lcm(n!,n!') = lcm(A000142(n),A068311(n)).

cp's OEIS Frontend

A359588 Dirichlet inverse of A083346.

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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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A085731 Greatest common divisor of n and its arithmetic derivative.

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A328571 Primorial base expansion of n converted into its prime product form, but with all nonzero digits replaced by 1's: a(n) = A007947(A276086(n)).

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A083347 Numbers k such that Sum(e/p: k=Product(p^e)) < 1.

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A083348 Numbers k such that r(k) = Sum(e/p: k = Product(p^e)) > 1.

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A189100 a(n) = lcm(n!,n!')/gcd(n!,n!'), where n!' is the arithmetic derivative of n! (A068311).

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A189102 Greatest common divisor of n! and its arithmetic derivative.

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