A018804 -id:A018804 - cp's OEIS frontend

A145388 Sum of (k,n)* for k=1,2,...,n, where (k,n)* is the greatest divisor of k which is a unitary divisor of n.

1, 3, 5, 7, 9, 15, 13, 15, 17, 27, 21, 35, 25, 39, 45, 31, 33, 51, 37, 63, 65, 63, 45, 75, 49, 75, 53, 91, 57, 135, 61, 63, 105, 99, 117, 119, 73, 111, 125, 135, 81, 195, 85, 147, 153, 135, 93, 155, 97, 147, 165, 175, 105, 159

Offset: 1

Laszlo Toth, Oct 10 2008

A unitary analog of Pillai's function A018804; another unitary analog of A018804 is A089912.
The sequence is the row sums of the following triangle of (k,n)* with rows n and columns 1 <= k <= n (_R. J. Mathar, Jun 01 2011):
  1;
  1,  2;
  1,  1,  3;
  1,  1,  1,  4;
  1,  1,  1,  1,  5;
  1,  2,  3,  2,  1,  6;
  1,  1,  1,  1,  1,  1,  7;
  1,  1,  1,  1,  1,  1,  1,  8;
  1,  1,  1,  1,  1,  1,  1,  1,  9;
  1,  2,  1,  2,  5,  2,  1,  2,  1, 10;
  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 11;
  1,  1,  3,  4,  1,  3,  1,  4,  3,  1,  1, 12;
  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 13;
  1,  2,  1,  2,  1,  2,  7,  2,  1,  2,  1,  2,  1, 14;
Sum_{k<=x} a(n) = Ax^2 log x + O(x^2) with A = Product(1 - 1/(p+1)^2) * 3/Pi^2 = 0.23584030... where the product is over the primes. That is, the average value of a(n) is A n log n. - Charles R Greathouse IV, Mar 21 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
S. Chen and W. Zhai, Reciprocals of the Gcd-Sum Functions, J. Int. Seq. 14 (2011) # 11.8.3.
László Tóth, The unitary analogue of Pillai's arithmetical function, Collectanea Mathematica 40:1 (1989), pp. 19-30.
László Tóth, The unitary analogue of Pillai's arithmetical function II, Notes Number Theory Discrete Math. 2 (1996), no 2, 40-46.
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009) 09.5.2.
László Tóth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010) # 10.8.1.

Cf. A000010, A018804, A034444, A047994, A089912.

A145388 := proc(n) option remember; local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; if nops(pf) = 1 then 2*n-1 ; else mul(procname(op(1,p)^op(2,p)),p=pf) ; end if; end if; end proc:
seq(A145388(n),n=1..70) ; # R. J. Mathar, Jan 07 2011

f[p_, e_] := 2*p^e - 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 29 2020 *)

a(n)=n=factor(n);prod(i=1,#n[,1],2*n[i,1]^n[i,2]-1) \\ Charles R Greathouse IV, Mar 21 2012

from math import prod
from sympy import factorint
def A145388(n): return prod((p**e<<1)-1 for p,e in factorint(n).items()) # Chai Wah Wu, Feb 13 2025

Multiplicative: a(p^e) = 2*p^e - 1 for every prime power p^e.
a(n) = Sum_{k=1..n} A034444(n/gcd(n,k)) = Sum_{d|n} A000010(d) * A034444(d). - Daniel Suteu, May 26 2019
a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * uphi(n/d), where uphi is A047994. - Amiram Eldar, May 29 2020
a(n) = Sum_{d|n} abs(A023900(d))*n/d. Verified for the first 10000 terms. - Mats Granvik, Feb 13 2021

A348981 a(n) = Sum_{d|n} phi(n/d) * A322582(d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 1, 12, 7, 11, 1, 24, 1, 15, 13, 32, 1, 35, 1, 40, 17, 23, 1, 68, 13, 27, 35, 56, 1, 71, 1, 80, 25, 35, 21, 112, 1, 39, 29, 116, 1, 99, 1, 88, 77, 47, 1, 176, 19, 91, 37, 104, 1, 151, 29, 164, 41, 59, 1, 232, 1, 63, 105, 192, 33, 155, 1, 136, 49, 159, 1, 308, 1, 75, 117, 152, 33, 183, 1, 304, 151

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of Euler phi (A000010) with A322582.

Möbius transform of A348980.

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

Cf. A000010, A003958, A008683, A018804, A322582, A348980 (Inverse Möbius transform), A348981, A348982, A348983, A349131.

Cf. also A347131, A349141.

f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#]) * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348981(n) = sumdiv(n,d,A322582(n/d)*eulerphi(d));

a(n) = Sum_{d|n} A000010(n/d) * A322582(d).
a(n) = Sum_{d|n} A008683(n/d) * A348980(d).
a(n) = Sum_{k=1..n} A322582(gcd(n,k)).
For all n >= 1, a(n) <= A347131(n) <= A349141(n).
a(n) = A018804(n) - A349131(n). - Antti Karttunen, Nov 14 2021

A349131 a(n) = Sum_{d|n} phi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and phi is Euler totient function.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 12, 8, 14, 16, 20, 16, 24, 24, 32, 16, 32, 28, 36, 32, 48, 40, 44, 32, 52, 48, 46, 48, 56, 64, 60, 32, 80, 64, 96, 56, 72, 72, 96, 64, 80, 96, 84, 80, 112, 88, 92, 64, 114, 104, 128, 96, 104, 92, 160, 96, 144, 112, 116, 128, 120, 120, 168, 64, 192, 160, 132, 128, 176, 192, 140, 112, 144, 144, 208

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A003958 with Euler totient function phi, A000010.

Möbius transform of A349130.

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

Cf. A000010, A003958, A018804, A348981, A349130 (inverse Möbius transform), A349132, A349171.

f[p_, e_] := (p - 1)*p^e - (p - 2)*(p - 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349131(n) = sumdiv(n,d,eulerphi(d)*A003958(n/d));

a(n) = Sum_{d|n} A000010(d) * A003958(n/d).
a(n) = Sum_{d|n} A008683(d) * A349130(n/d).
a(n) = Sum_{k=1..n} A003958(gcd(n, k)).
a(n) = A018804(n) - A348981(n).
For all n >= 1, a(n) <= A349171(n).
Multiplicative with a(p^e) = (p-1)*p^e - (p-2)*(p-1)^e. - Amiram Eldar, Nov 09 2021
Dirichlet g.f.: (zeta(s-1)/zeta(s)) / Product_{p prime} (1 - 1/p^(s-1) + 1/p^s). - Amiram Eldar, Dec 24 2023

A349141 a(n) = Sum_{d|n} phi(n/d) * A348507(d), where A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 6, 1, 9, 1, 26, 9, 13, 1, 44, 1, 17, 15, 98, 1, 57, 1, 68, 19, 25, 1, 176, 15, 29, 57, 92, 1, 105, 1, 342, 27, 37, 23, 252, 1, 41, 31, 280, 1, 141, 1, 140, 111, 49, 1, 636, 21, 125, 39, 164, 1, 309, 31, 384, 43, 61, 1, 480, 1, 65, 147, 1138, 35, 213, 1, 212, 51, 209, 1, 960, 1, 77, 155, 236, 35, 249, 1, 1028

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of Euler phi (A000010) with A348507.

Möbius transform of A349140.

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

Cf. A000010, A003959, A008683, A018804, A348507, A349140 (inverse Möbius transform), A349142, A349143, A349171.

Cf. also A347131, A348981.

f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (s[#] - #) * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349141(n) = sumdiv(n,d,eulerphi(d)*A348507(n/d));

a(n) = Sum_{d|n} A000010(n/d) * A348507(d).
a(n) = Sum_{d|n} A008683(n/d) * A349140(d).
a(n) = Sum_{k=1..n} A348507(gcd(n,k)).
For all n >= 1, a(n) >= A347131(n) >= A348981(n).
a(n) = A349171(n) - A018804(n). - Antti Karttunen, Nov 14 2021

A349171 a(n) = Sum_{d|n} phi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and phi is Euler totient function.

Original entry on oeis.org

1, 4, 6, 14, 10, 24, 14, 46, 30, 40, 22, 84, 26, 56, 60, 146, 34, 120, 38, 140, 84, 88, 46, 276, 80, 104, 138, 196, 58, 240, 62, 454, 132, 136, 140, 420, 74, 152, 156, 460, 82, 336, 86, 308, 300, 184, 94, 876, 154, 320, 204, 364, 106, 552, 220, 644, 228, 232, 118, 840, 122, 248, 420, 1394, 260, 528, 134, 476, 276

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A003959 with Euler totient function phi, A000010.

Möbius transform of A349170.

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

Cf. A000010, A003959, A018804, A349141, A349170 (inverse Möbius transform), A349172, A349131.

f[p_, e_] := p*(p + 1)^e - (p - 1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349171(n) = sumdiv(n,d,eulerphi(d)*A003959(n/d));

a(n) = Sum_{d|n} A000010(d) * A003959(n/d).
a(n) = Sum_{d|n} A008683(d) * A349170(n/d).
a(n) = Sum_{k=1..n} A003959(gcd(n, k)).
a(n) = A018804(n) + A349141(n).
For all n >= 1, a(n) >= A349131(n).
Multiplicative with a(p^e) = p*(p+1)^e - (p-1)*p^e. - Amiram Eldar, Nov 09 2021

A360428 Inverse Mobius transformation of A338164.

Original entry on oeis.org

1, 7, 17, 40, 49, 119, 97, 208, 225, 343, 241, 680, 337, 679, 833, 1024, 577, 1575, 721, 1960, 1649, 1687, 1057, 3536, 1825, 2359, 2673, 3880, 1681, 5831, 1921, 4864, 4097, 4039, 4753, 9000, 2737, 5047, 5729, 10192, 3361, 11543, 3697, 9640, 11025, 7399, 4417, 17408, 7105, 12775

Offset: 1

R. J. Mathar, Feb 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001620, A007434, A008683, A018804, A069097, A338164.

A360428 := proc(n)
    add(numtheory[mobius](n/d)*numtheory[tau](d)*d^2,d=numtheory[divisors](n)) ;
end proc:

f[p_, e_] := (e + 1 - e/p^2)*p^(2*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 09 2023 *)

a(n) = Sum_{d|n} A008683(n/d)*A000005(d)*d^2.
Dirichlet convolution of A034714 and A008683.
Dirichlet g.f.: zeta^2(s-2)/zeta(s).
From Amiram Eldar, Feb 09 2023: (Start)
Multiplicative with a(p^e) = (e + 1 - e/p^2)*p^(2*e).
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1/3 - zeta'(3)/zeta(3)) * n^3 / (3*zeta(3)), where gamma is Euler's constant (A001620). (End)
From Peter Bala, Jan 16 2024: (Start)
a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^2. Cf. A069097.
a(n) = Sum_{d divides n} d^2 * J_2(n/d), where J_2(n) = A007434(n). (End)

A299152 Denominators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 03 2018

			Sequence begins: 1, 1, 2, 7/2, 8, 14, 32, 121/2, 126, 248, 512, 1003, 2048, 4064, 8176, 130539/8, 32768.

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

Cf. A000005, A000010, A000740, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299119, A299149, A299151.

nn=50;
sys=Table[2^(n-1)==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Denominator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]

up_to = 65537;
prepareA299151perA299152(up_to) = { my(vmemo = vector(up_to)); for(n=1,up_to, vmemo[n] = if(1==n,n,(2^(n-1)-sumdiv(n,d,if((d>1)&&(dA299152 = prepareA299151perA299152(up_to);
A299151perA299152(n) = v299151perA299152[n];
\\ Or without memoization as:
A299151perA299152(n) = if(1==n,n,(2^(n-1)-sumdiv(n,d,if((d>1)&&(dA299151perA299152(d)*A299151perA299152(n/d),0)))/2);
A299152(n) = denominator(A299151perA299152(n)); \\ Antti Karttunen, Jul 29 2018

More terms from Antti Karttunen, Jul 29 2018

A327251 Expansion of Sum_{k>=1} psi(k) * x^k / (1 - x^k)^2, where psi = A001615.

Original entry on oeis.org

1, 5, 7, 16, 11, 35, 15, 44, 33, 55, 23, 112, 27, 75, 77, 112, 35, 165, 39, 176, 105, 115, 47, 308, 85, 135, 135, 240, 59, 385, 63, 272, 161, 175, 165, 528, 75, 195, 189, 484, 83, 525, 87, 368, 363, 235, 95, 784, 161, 425, 245, 432, 107, 675, 253, 660, 273

Offset: 1

Ilya Gutkovskiy, Sep 15 2019

Inverse Moebius transform of A322577.

Dirichlet convolution of A001615 with A000027.

Cf. A000027, A001615, A018804, A060648, A322577, A347127.

nmax = 57; CoefficientList[Series[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k] x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := p^(e - 1)*((p + 1)*e + p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)

mypsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = sumdiv(n, d, mypsi(n/d)*d); \\ Michel Marcus, Sep 15 2019

a(n) = Sum_{d|n} psi(n/d) * d.
a(p) = 2*p + 1, where p is prime.
Multiplicative with a(p^e) = p^(e-1)*((p+1)*e + p). - Antti Karttunen, Aug 24 2021

A349569 Dirichlet convolution of A000027 (identity function) with A349452 (Dirichlet inverse of A011782, 2^(n-1)).

Original entry on oeis.org

1, 0, -1, -4, -11, -24, -57, -112, -243, -480, -1013, -1964, -4083, -8064, -16309, -32496, -65519, -130440, -262125, -523156, -1048263, -2095104, -4194281, -8383760, -16777015, -33546240, -67107609, -134200860, -268435427, -536835096, -1073741793, -2147417216, -4294962187, -8589803520, -17179867533, -34359463812

Offset: 1

Antti Karttunen, Nov 22 2021

sign

Dirichlet convolution with A034729 gives sigma, A000203, and convolution with A034738 gives A018804.

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A000027, A011782, A349452, A349570 (Dirichlet inverse).

Cf. also A000203, A018804, A034729, A034738, A349563, A349565, A349567.

s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#]*2^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, # * s[n/#] &]; Array[a, 36] (* Amiram Eldar, Nov 22 2021 *)

A011782(n) = (2^(n-1));
memoA349452 = Map();
A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(dA011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));
A349569(n) = sumdiv(n,d,d * A349452(n/d));

a(n) = Sum_{d|n} d * A349452(n/d).

A368742 a(n) = Sum_{k = 1..n} gcd(6*k, n).

Original entry on oeis.org

1, 4, 9, 12, 9, 36, 13, 32, 45, 36, 21, 108, 25, 52, 81, 80, 33, 180, 37, 108, 117, 84, 45, 288, 65, 100, 189, 156, 57, 324, 61, 192, 189, 132, 117, 540, 73, 148, 225, 288, 81, 468, 85, 252, 405, 180, 93, 720, 133, 260, 297, 300, 105, 756, 189, 416, 333, 228, 117, 972, 121, 244, 585, 448

Offset: 1

Peter Bala, Jan 08 2024

a(n) equals the number of solutions to the congruence 6*x*y == 0 (mod n) for 1 <= x, y <= n.

			a(4) = 12: each of the 16 pairs (x, y), 1 <= x, y <= 4, is a solution to the congruence 6*x*y == 0 (mod 4) except for the 4 pairs (1, 1) , (1, 3), (3, 1) and (3, 3) with both x and y odd.

Cf. A000010, A018804, A344372, A368736 - A368741.

seq(add(gcd(6*k, n), k = 1..n), n = 1..70);
# alternative faster program for large n
with(numtheory): seq(add(gcd(6,d)*phi(d)*n/d, d in divisors(n)), n = 1..70);

Table[Sum[GCD[6*k, n], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 12 2024 *)

a(6*n) = 36*A018804(n); a(6*n+2) = 4*A018804(3*n+1);
a(6*n+3) = 9*A018804(2*n+1); a(6*n+4) = 4*A018804(3*n+2);
a(6*n+r) = A018804(6*n+r) for r = 1 and 5.
Define a_m(n) = Sum_{k = 1..n} gcd(m*k, n). Then
a(n) = a_2(n) * a_3(n) / a_1(n) = A344372(n) * A368737(n) / A018804(n).
a(n) = Sum_{d divides n} gcd(6, d)*phi(d)*n/d, where phi(n) = A000010(n).
Multiplicative: a(2^k) = (k + 1)*2^k, a(3^k) = (2*k + 1)*3^k, and for prime p > 3, a(p^k) = (k + 1)*p^k - k*p^(k-1).
Dirichlet g.f.: ( 1 + 3/3^s)/((1 - 1/2^s)*(1 - 1/3^s)) * zeta(s-1)^2/zeta(s).
Sum_{k=1..n} a(k) ~ n^2 * (6*log(n) - 3 + 12*gamma - 2*log(2) - 9*log(3)/4 - 36*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 12 2024

cp's OEIS Frontend

A145388 Sum of (k,n)* for k=1,2,...,n, where (k,n)* is the greatest divisor of k which is a unitary divisor of n.

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A348981 a(n) = Sum_{d|n} phi(n/d) * A322582(d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

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A349131 a(n) = Sum_{d|n} phi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and phi is Euler totient function.

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A349141 a(n) = Sum_{d|n} phi(n/d) * A348507(d), where A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

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A349171 a(n) = Sum_{d|n} phi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and phi is Euler totient function.

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A360428 Inverse Mobius transformation of A338164.

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A299152 Denominators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

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A327251 Expansion of Sum_{k>=1} psi(k) * x^k / (1 - x^k)^2, where psi = A001615.