A001615 -id:A001615 - cp's OEIS frontend

A307868 Decimal expansion of the asymptotic mean of phi(k)/psi(k), where phi(k) is Euler totient function (A000010) and psi(k) is Dedekind psi function (A001615).

4, 7, 1, 6, 8, 0, 6, 1, 3, 6, 1, 2, 9, 9, 7, 8, 6, 8, 0, 7, 5, 2, 3, 5, 6, 3, 3, 0, 8, 0, 4, 8, 2, 0, 8, 7, 4, 2, 5, 9, 2, 6, 3, 8, 2, 0, 0, 6, 9, 8, 6, 8, 8, 3, 6, 3, 5, 7, 3, 7, 2, 5, 5, 4, 1, 7, 7, 3, 2, 1, 1, 6, 7, 5, 9, 6, 8, 2, 7, 4, 4, 0, 9, 6, 2, 1, 0, 0, 2, 7, 3, 7, 6, 9, 4, 9, 0, 2, 3, 0, 3, 1, 3, 0, 1, 1

Offset: 0

Amiram Eldar, May 02 2019

Also, the asymptotic mean of A162511. - Amiram Eldar, Sep 18 2022

			0.47168061361299786807523563308048208742592638200698...

V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link.

Cf. A000010, A001615, A013661, A065472, A071974, A162511, A173557.

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-2, 1, 2}, {0, -4, 6}, m]; RealDigits[(2/3) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

prodeulerrat(1 - 2/(p*(p+1))) \\ Vaclav Kotesovec, Sep 19 2020

Equals lim_{m->oo} (1/m)*Sum_{k=1..m} phi(k)/psi(k).
Equals Product_{p prime} (1 - 2/(p * (p+1))).
Equals A065472 / zeta(2). - Amiram Eldar, Sep 18 2022

More digits from Vaclav Kotesovec, Sep 19 2020

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).

A323363 Dirichlet inverse of Dedekind's psi, A001615.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 13 2019

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

Cf. A048250 (absolute values).

Cf. A001615, A008836, A323364, A323372.

psi[n_] := If[n == 1, 1, n Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
a[n_] := a[n] = If[n == 1, 1, -Sum[psi[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 75] (* Jean-François Alcover, Feb 15 2020 *)
f[p_, e_] := (-1)^e * (p + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
A323363(n) = if(1==n,1,-sumdiv(n,d,if(dA001615(n/d)*A323363(d),0)));

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} psi(k) * A(x^k). - Ilya Gutkovskiy, Sep 04 2019
From Amiram Eldar, Oct 14 2020: (Start)
Multiplicative with a(p^e) = (-1)^e * (p+1).
a(n) = A008836(n) * A048250(n). (End)
Dirichlet g.f.: zeta(2*s)/(zeta(s-1)*zeta(s)). - Amiram Eldar, Dec 05 2022

A347385 Dedekind psi function applied to the odd part of n: a(n) = A001615(A000265(n)).

Original entry on oeis.org

1, 1, 4, 1, 6, 4, 8, 1, 12, 6, 12, 4, 14, 8, 24, 1, 18, 12, 20, 6, 32, 12, 24, 4, 30, 14, 36, 8, 30, 24, 32, 1, 48, 18, 48, 12, 38, 20, 56, 6, 42, 32, 44, 12, 72, 24, 48, 4, 56, 30, 72, 14, 54, 36, 72, 8, 80, 30, 60, 24, 62, 32, 96, 1, 84, 48, 68, 18, 96, 48, 72, 12, 74, 38, 120, 20, 96, 56, 80, 6, 108, 42, 84, 32, 108

Offset: 1

Antti Karttunen, Aug 31 2021

Coincides with A000593 on A122132.

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

Cf. A000265, A001615, A122132, A185199, A206787, A336651, A347386, A347387, A351035.

Cf. also A000593, A053575.

f[p_, e_] := If[p == 2, 1, (p + 1)*p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2021 *)

A347385(n) = if(1==n,n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));

Multiplicative with a(2^e) = 1, a(p^e) = (p+1)*p^(e-1) for all odd primes p.
a(n) = A001615(A000265(n)).
a(n) = A206787(n) * A336651(n). - Antti Karttunen, Feb 11 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). - Amiram Eldar, Nov 19 2022
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s))*(4^s-2^(s+1))/(4^s-1). - Amiram Eldar, Jan 04 2023

A357817 Partial alternating sums of the Dedekind psi function (A001615): a(n) = Sum_{k=1..n} (-1)^(k+1) * psi(k).

Original entry on oeis.org

1, -2, 2, -4, 2, -10, -2, -14, -2, -20, -8, -32, -18, -42, -18, -42, -24, -60, -40, -76, -44, -80, -56, -104, -74, -116, -80, -128, -98, -170, -138, -186, -138, -192, -144, -216, -178, -238, -182, -254, -212, -308, -264, -336, -264, -336, -288, -384, -328, -418

Offset: 1

Amiram Eldar, Oct 14 2022

sign

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A001615, A173290.

Similar sequences: A068762, A068773, A307704.

psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Accumulate[Array[(-1)^(# + 1)*psi[#] &, 50]]

f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = sum(k=1, n, (-1)^(k+1) * f(k)); \\ Michel Marcus, Oct 15 2022

a(n) = -(3/(2*Pi^3)) * n^2 + O(n * log(n)^(2/3)) (Tóth, 2017).

A173290 Partial sums of A001615.

Original entry on oeis.org

1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, 132, 156, 180, 204, 222, 258, 278, 314, 346, 382, 406, 454, 484, 526, 562, 610, 640, 712, 744, 792, 840, 894, 942, 1014, 1052, 1112, 1168, 1240, 1282, 1378, 1422, 1494, 1566, 1638, 1686, 1782, 1838, 1928, 2000, 2084

Offset: 1

Jonathan Vos Post, Feb 15 2010

nonn

a(n) is even for n >= 2. - Jianing Song, Nov 24 2018

W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88; http://scientificadvances.co.in; DOI: http://dx.doi.org/10.18642/jantaa_7100121599

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88.

Cf. A001615, A063659.
Cf. A082020.
Cf. A175836 (partial products of the Dedekind psi function).

[(&+[MoebiusMu(k)^2*Floor(n/k)*Floor(1 + n/k): k in [1..n]])/2: n in [1..60]]; // G. C. Greubel, Nov 23 2018

with(numtheory): a:=n->(1/2)*add(mobius(k)^2*floor(n/k)*floor(1+n/k),k=1..n); seq(a(n),n=1..55); # Muniru A Asiru, Nov 24 2018

Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k], {k,1,n}], {n,60}] (* G. C. Greubel, Nov 23 2018 *)
psi[n_] := If[n==1, 1, n*Times@@(1 + 1/FactorInteger[n][[;;,1]])]; Accumulate[Array[psi, 50]] (* Amiram Eldar, Nov 23 2018 *)

S(n) = sum(k=1, sqrtint(n), moebius(k)*(n\(k*k))); \\ see: A013928
a(n) = sum(k=1, sqrtint(n), k*(k+1) * (S(n\k) - S(n\(k+1))))/2 + sum(k=1, n\(1+sqrtint(n)), moebius(k)^2*(n\k)*(1+n\k))/2; \\ Daniel Suteu, Nov 23 2018

def A173290(n) :
    return add(k*mul(1+1/p for p in prime_divisors(k)) for k in (1..n))
[A173290(n) for n in (1..52)]  # Peter Luschny, Jun 10 2012

a(n) = Sum_{i=1..n} A001615(i) = Sum_{i=1..n} (n * Product_{p|n, p prime} (1 + 1/p)).
a(n) = 15*n^2/(2*Pi^2) + O(n*log(n)). - Enrique Pérez Herrero, Jan 14 2012
a(n) = Sum_{i=1..n} A063659(i) * floor(n/i). - Enrique Pérez Herrero, Feb 23 2013
a(n) = (1/2)*Sum_{k=1..n} mu(k)^2 * floor(n/k) * floor(1+n/k), where mu(k) is the Moebius function. - Daniel Suteu, Nov 19 2018
a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (A013928(1+floor(n/k)) - A013928(1+floor(n/(k+1)))) + Sum_{k=1..floor(n/(1+floor(sqrt(n))))} mu(k)^2 * floor(n/k) * floor(1+n/k))/2. - Daniel Suteu, Nov 23 2018

A175836 a(n) = Product_{i=1..n} psi(i) where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 12, 72, 432, 5184, 41472, 497664, 5971968, 107495424, 1289945088, 30958682112, 433421549568, 10402117189632, 249650812551168, 5991619501228032, 107849151022104576, 3882569436795764736

Offset: 1

Enrique Pérez Herrero, Sep 18 2010

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = A060648(gcd(i,j)) for 1 <= i,j <= n, note that A060648 is the Inverse Möbius transform of A001615. - Enrique Pérez Herrero, Aug 12 2011

Charles R Greathouse IV, Table of n, a(n) for n = 1..423
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49
Eric Weisstein's World of Mathematics, Le Paige's Theorem
Index to divisibility sequences

Cf. A001615, A001088, A059381, A059382, A059383, A059384, A238498.

Cf. A239672.

a175836 n = a175836_list !! (n-1)
a175836_list = scanl1 (*) a001615_list
-- Reinhard Zumkeller, Mar 01 2014

A175836 := proc(n) option remember; local p; `if`(n<2,1, n*mul(1+1/p,p=factorset(n))*A175836(n-1)) end: # Peter Luschny, Feb 28 2014

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n];
DedekindPsi[n_]:=JordanTotient[n,2]/EulerPhi[n];
A175836[n_]:=Times@@DedekindPsi/@Range[n]

a=direuler(p=2, 100, (1+X)/(1-p*X));for(i=2,#a,a[i]*=a[i-1]);a
\\ Charles R Greathouse IV, Jul 29 2011

a(n) = A059381(n)/A001088(n).

A306927 a(n) = A001615(n) - n.

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 4, 3, 8, 1, 12, 1, 10, 9, 8, 1, 18, 1, 16, 11, 14, 1, 24, 5, 16, 9, 20, 1, 42, 1, 16, 15, 20, 13, 36, 1, 22, 17, 32, 1, 54, 1, 28, 27, 26, 1, 48, 7, 40, 21, 32, 1, 54, 17, 40, 23, 32, 1, 84, 1, 34, 33, 32, 19, 78, 1, 40, 27, 74, 1, 72

Offset: 1

Torlach Rush, Mar 16 2019

Analogous to A051953.
a(n) = A051953(n) if n is an element of A000961.
a(n) > A051953(n) if n is an element of A024619.
The sum of the proper divisors d of n such that n/d is squarefree. - Amiram Eldar, Sep 06 2019

			0 is a term because A001615(1) - 1 = 0.
1 is a term because A001615(2) - 2 = 1.
3 is a term because A001615(9) - 9 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Dedekind psi function.

Cf. A000961, A001615, A024619, A051953, A082020.

a[1] = 0; a[n_] := n * (Times @@ (1 + 1/FactorInteger[n][[;; , 1]]) - 1); Array[a, 100] (* Amiram Eldar, Sep 06 2019 *)

a(n) = n*(sumdivmult(n, d, issquarefree(d)/d) - 1); \\ Michel Marcus, Mar 18 2019

a(n) = A001615(n) - n.
a(n) = Sum_{d|n, dAmiram Eldar, Sep 06 2019
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 - 1 = 0.519817... . - Amiram Eldar, Dec 08 2023

A344696 a(n) = sigma(n) / gcd(sigma(n), A001615(n)).

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 5, 31, 1, 10, 7, 1, 1, 1, 21, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 10, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 65, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 21, 1, 57, 13

Offset: 1

Antti Karttunen, 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) = 35, although a(4) = 7 and a(27) = 10. See A344702.

Antti Karttunen, Table of n, a(n) for n = 1..10404
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A001615, A005117 (positions of ones), A344695, A344697, A344698, A344702.

Cf. also A344756.

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
A344696(n) = { my(u=sigma(n)); (u/gcd(u,A001615(n))); };

a(n) = A000203(n) / A344695(n).

A344697 a(n) = A001615(n) / gcd(sigma(n), A001615(n)).

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 1, 4, 12, 1, 1, 6, 1, 1, 1, 24, 1, 12, 1, 6, 1, 1, 1, 4, 30, 1, 9, 6, 1, 1, 1, 16, 1, 1, 1, 72, 1, 1, 1, 4, 1, 1, 1, 6, 12, 1, 1, 24, 56, 30, 1, 6, 1, 9, 1, 4, 1, 1, 1, 6, 1, 1, 12, 96, 1, 1, 1, 6, 1, 1, 1, 48, 1, 1, 30, 6, 1, 1, 1, 24, 108, 1, 1, 6, 1, 1, 1, 4, 1, 12, 1, 6, 1, 1, 1, 16, 1, 56, 12, 180

Offset: 1

Antti Karttunen, 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) = 27, although a(4) = 6 and a(27) = 9. See A344702.

Antti Karttunen, Table of n, a(n) for n = 1..10404
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A001615, A005117, A344695, A344696, A344699, A344702.

Cf. also A344757.

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
A344697(n) = { my(u=A001615(n)); (u/gcd(u,sigma(n))); };

a(n) = A001615(n) / A344695(n).

cp's OEIS Frontend

A307868 Decimal expansion of the asymptotic mean of phi(k)/psi(k), where phi(k) is Euler totient function (A000010) and psi(k) is Dedekind psi function (A001615).

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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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A323363 Dirichlet inverse of Dedekind's psi, A001615.

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A347385 Dedekind psi function applied to the odd part of n: a(n) = A001615(A000265(n)).

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A357817 Partial alternating sums of the Dedekind psi function (A001615): a(n) = Sum_{k=1..n} (-1)^(k+1) * psi(k).

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A173290 Partial sums of A001615.

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A175836 a(n) = Product_{i=1..n} psi(i) where psi is the Dedekind psi function (A001615).

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