A189100 -id:A189100 - cp's OEIS frontend

A085731 Greatest common divisor of n and its arithmetic derivative.

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)

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)).

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)).

A188901 Integers in the sequences (arithmetic derivative of k) divided by k.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 3, 2, 5, 4, 3, 1, 6, 5, 4, 2, 7, 3, 6, 5, 3, 8, 4, 2, 7, 6, 4, 9, 5, 3, 8, 4, 7, 5, 1, 10, 6, 4, 9, 5, 3, 8, 6, 2, 11, 7, 5, 10, 6, 4, 2, 9, 7, 3, 5, 12, 8, 6, 2, 11, 7, 5, 3, 10, 8, 4, 6, 4, 13, 9, 7, 3, 12, 8, 6, 4, 11

Offset: 1

Giorgio Balzarotti, Apr 16 2011

nonn

Integers in the sequence (k'/k) = A003415(k)/k. See A072873 for the values of k where k'/k is an integer.

			1' = 0, 0/1 = 0 -> a(1) = 0;
4' = 4 ,4/4 = 1 -> a(2) = 1;
16' = 32, 32/16 = 2 -> a(3) = 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 2500 terms from Nathaniel Johnston)

Cf. A003415, A068311, A072873, A083346, A083346, A085731, A086130, A189100, A189102.

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A085731 Greatest common divisor of n and its arithmetic derivative.

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

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

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A188901 Integers in the sequences (arithmetic derivative of k) divided by k.

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