A190116 -id:A190116 - 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)

A327861 Number of divisors d of n for which A003415(d)*d is equal to n, where A003415(x) gives the arithmetic derivative of x.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Antti Karttunen, Sep 28 2019

nonn

Number of times n occurs in A190116.

			a(4153248)=2 as out of 192 divisors of 4153248, only 1368 and 2277 are such that 1368 * A003415(1368) = 2277 * A003415(2277) = 4153248.

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

Cf. A003415, A190116.

Cf. also A327153, A327166, A327169.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A327861(n) = sumdiv(n,d,(d*A003415(d) == n));

A348279 a(n) = Sum_{d|n} d*d', where d' is the arithmetic derivative of d (A003415).

Original entry on oeis.org

0, 2, 3, 18, 5, 35, 7, 114, 57, 77, 11, 243, 13, 135, 128, 626, 17, 467, 19, 573, 220, 299, 23, 1395, 255, 405, 786, 1047, 29, 1160, 31, 3186, 476, 665, 432, 2835, 37, 819, 640, 3389, 41, 2100, 43, 2427, 1937, 1175, 47, 7283, 693, 2577, 1040, 3333, 53, 5570, 896, 6295, 1276

Offset: 1

Wesley Ivan Hurt, Oct 09 2021

nonn

			a(4) = 18; a(4) = 1*1' + 2*2' + 4*4' = 1*0 + 2*1 + 4*4 = 18.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003415 (arithmetic derivative).
Inverse Möbius transform of A190116.
Cf. also A347130.

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, d*ad(d)); \\ Michel Marcus, Oct 10 2021

a(p) = p for primes p since we have a(p) = 1*1' + p*p' = 1*0 + p*1 = p.

a(n) = Sum_{d|n} A190116(d). - Antti Karttunen, Dec 07 2021

A190117 a(n) = Sum_{k=1..n} k*k', where n' is the arithmetic derivative of n.

Original entry on oeis.org

0, 2, 5, 21, 26, 56, 63, 159, 213, 283, 294, 486, 499, 625, 745, 1257, 1274, 1652, 1671, 2151, 2361, 2647, 2670, 3726, 3976, 4366, 5095, 5991, 6020, 6950, 6981, 9541, 10003, 10649, 11069, 13229, 13266, 14064, 14688, 17408, 17449, 19171, 19214, 21326, 23081, 24231, 24278, 29654, 30340, 32590

Offset: 1

Giorgio Balzarotti, May 04 2011

nonn

			1*1' + 2*2' + 3*3' = 0 + 2 + 3 = 5 -> a(3) = 5.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Partial sums of A190116.

Cf. A003415, A136141.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
seq(add(der(i)*i,i=1..n),n=1..50);

A003415[n_]:= If[Abs@n < 2, 0, n Total[#2/#1 & @@@FactorInteger[Abs@n]]];
Table[Sum[k*A003415[k], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Dec 29 2017 *)

a(n) ~ c * n^3 / 3, where c = Sum_{p prime} 1/(p*(p-1)) = A136141. - Amiram Eldar, Jun 22 2025

A359331 Nonprime numbers k for which k*k' is a palindrome, where k' is the arithmetic derivative of k (A003415).

Original entry on oeis.org

1, 34, 44, 49, 121, 476, 524, 533, 1808, 6797, 7326, 10016, 10201, 10403, 10817, 16019, 17831, 26322, 33898, 55198, 57247, 74711, 87241, 131395, 148753, 156029, 239593, 240021, 289831, 295022, 423758, 441691, 595777, 725754, 900009, 2568543, 2910271, 2981619

Offset: 1

Marius A. Burtea, Jan 29 2023

			1*1' = 1*0 = 0, so 1 is a term.
34*34' = 34*19 = 646, so 34 is a term.
49*49' = 49*14 = 686, so 49 is a term.

Cf. A002113, A003415, A018252, A190116.

f:=func; pal:=func; [n:n in [1..3000000]|not IsPrime(n) and pal(n*Floor(f(n)))];

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
q:= n-> not isprime(n) and StringTools[IsPalindrome](""||(n*d(n))):
select(q, [$1..3000000])[];  # Alois P. Heinz, Jan 29 2023

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[3*10^6], ! PrimeQ[#] && PalindromeQ[# * d[#]] &] (* Amiram Eldar, Jan 29 2023 *)

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A085731 Greatest common divisor of n and its arithmetic derivative.

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A327861 Number of divisors d of n for which A003415(d)*d is equal to n, where A003415(x) gives the arithmetic derivative of x.

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A348279 a(n) = Sum_{d|n} d*d', where d' is the arithmetic derivative of d (A003415).

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A190117 a(n) = Sum_{k=1..n} k*k', where n' is the arithmetic derivative of n.

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A359331 Nonprime numbers k for which k*k' is a palindrome, where k' is the arithmetic derivative of k (A003415).

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