A340070 -id:A340070 - cp's OEIS frontend

A346469 a(n) = A340070(A276086(n)).

0, 1, 1, 5, 3, 3, 1, 7, 8, 31, 3, 3, 5, 5, 5, 5, 120, 15, 25, 25, 50, 25, 75, 75, 125, 125, 125, 125, 750, 375, 1, 9, 10, 41, 3, 3, 12, 59, 71, 247, 3, 3, 5, 5, 5, 5, 15, 15, 50, 25, 25, 25, 75, 75, 375, 125, 125, 125, 375, 375, 7, 7, 7, 7, 210, 21, 7, 7, 7, 7, 21, 21, 420, 35, 35, 35, 7455, 105, 175, 175, 175, 3325

Offset: 0

Antti Karttunen, Jul 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A069359, A276086, A327860, A329029, A340070, A345930.

Cf. also A346471.

A346469(n) = { my(s=0, t=0, m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p^e); s += (1/p); t += (e/p)); n = n\p; p = nextprime(1+p)); (gcd(s,t)*m); };

a(n) = A340070(A276086(n)) = gcd(A327860(n), A329029(n)).

For n >= 1, a(n) = A327860(n) / A345930(n).

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)

A344695 a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32, 108

Offset: 1

Antti Karttunen and Peter Munn, May 26 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 8, although a(4) = 1 and a(27) = 4. See A344702.
A more specific property holds: for prime p that does not divide n, a(p*n) = a(p) * a(n). In particular, on squarefree numbers (A005117) this sequence coincides with sigma and psi, which are multiplicative.
If prime p divides the squarefree part of n then p+1 divides a(n). (For example, 20 has square part 4 and squarefree part 5, so 5+1 divides a(20) = 6.) So a(n) = 1 only if n is square. The first square n with a(n) > 1 is a(196) = 21. See A344703.
Conjecture: the set of primes that appear in the sequence is A065091 (the odd primes). 5 does not appear as a term until a(366025) = 5, where 366025 = 5^2 * 11^4. At this point, the missing numbers less than 22 are 2, 10 and 17. 17 appears at the latest by a(17^2 * 103^16) = 17.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A001615, A005117, A244963, A344696, A344697, A344702, A344703 (numbers k for which a(k^2) > 1).
Cf. also A048250, A291750, A291751, A340070, A323363.
Subsets of range: A008864, A065091 (conjectured).

Table[GCD[DivisorSigma[1,n],DivisorSum[n,MoebiusMu[n/#]^2*#&]],{n,100}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344695(n) = gcd(sigma(n), A001615(n));
(Python 3.8+)
from math import prod, gcd
from sympy import primefactors, divisor_sigma
def A001615(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist)
def A344695(n): return gcd(A001615(n),divisor_sigma(n)) # Chai Wah Wu, Jun 03 2021

a(n) = gcd(A000203(n), A001615(n)).
For prime p, a(p^e) = (p+1)^(e mod 2).
For prime p with gcd(p, n) = 1, a(p*n) = a(p) * a(n).
a(A007913(n)) | a(n).
a(n) = gcd(A000203(n), A244963(n)) = gcd(A001615(n), A244963(n)).
a(n) = A000203(n) / A344696(n).
a(n) = A001615(n) / A344697(n).

A344756 a(n) = A003415(n) / gcd(A003415(n), A069359(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 8, 1, 1, 1, 4, 1, 7, 1, 12, 1, 1, 1, 11, 2, 1, 3, 16, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 17, 1, 1, 1, 24, 13, 1, 1, 14, 2, 9, 1, 28, 1, 9, 1, 23, 1, 1, 1, 46, 1, 1, 17, 6, 1, 1, 1, 36, 1, 1, 1, 13, 1, 1, 11, 40, 1, 1, 1, 22, 4, 1, 1, 62, 1, 1, 1, 35, 1, 41, 1, 48, 1, 1, 1, 17, 1, 11, 25

Offset: 2

Antti Karttunen, May 28 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 2..12012
Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

Cf. A003415, A005117 (for n > 1 gives the positions of ones), A069359, A340070, A344757.

Cf. also A344696.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A344756(n) = { my(u=A003415(n)); (u/gcd(u,A069359(n))); };

a(n) = A003415(n) / A340070(n) = A003415(n) / gcd(A003415(n), A069359(n)).

A344757 a(n) = A069359(n) / gcd(A003415(n), A069359(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 5, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 13, 8, 1, 1, 5, 1, 7, 1, 15, 1, 5, 1, 9, 1, 1, 1, 31, 1, 1, 10, 1, 1, 1, 1, 19, 1, 1, 1, 5, 1, 1, 8, 21, 1, 1, 1, 7, 1, 1, 1, 41, 1, 1, 1, 13, 1, 31, 1, 25, 1, 1, 1, 5, 1, 9, 14, 1, 1, 1, 1, 15

Offset: 2

Antti Karttunen, May 28 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 2..12012
Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

Cf. A003415, A069359, A340070, A344756.

Cf. also A344697.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A344757(n) = { my(u=A069359(n)); (u/gcd(u,A003415(n))); };

a(n) = A069359(n) / A340070(n) = A069359(n) / gcd(A003415(n), A069359(n)).

cp's OEIS Frontend

A346469 a(n) = A340070(A276086(n)).

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A085731 Greatest common divisor of n and its arithmetic derivative.

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A344695 a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615.

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A344756 a(n) = A003415(n) / gcd(A003415(n), A069359(n)).

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A344757 a(n) = A069359(n) / gcd(A003415(n), A069359(n)).

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