Laszlo Toth - cp's OEIS frontend

A280184 Number of cyclic subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

1, 16, 41, 136, 157, 656, 401, 1096, 1121, 2512, 1465, 5576, 2381, 6416, 6437, 8776, 5221, 17936, 7241, 21352, 16441, 23440, 12721, 44936, 19657, 38096, 30281, 54536, 25261, 102992, 30785, 70216, 60065, 83536, 62957, 152456, 52061, 115856, 97621, 172072, 70645, 263056, 81401, 199240, 175997, 203536, 106081, 359816, 137601, 314512

Offset: 1

Laszlo Toth, Dec 28 2016

Inverse Moebius transform of A160891. - Seiichi Manyama, May 12 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013662, A060648, A064969, A160891, A280162, A344219, A344302, A344303.

with(numtheory):
# define Jordan totient function J(r,n)
J(r,n) := add(d^r*mobius(n/d), d in divisors(n)):
seq(add(J(4,d)/phi(d), d in divisors(n)), n = 1..50); # Peter Bala, Jan 23 2024

a[n_] := With[{dd = Divisors[n]}, Sum[Times @@ EulerPhi @ {x, y, z, t} / EulerPhi[LCM[x, y, z, t]], {x, dd}, {y, dd}, {z, dd}, {t, dd}]];
Array[a, 50] (* Jean-François Alcover, Sep 28 2018 *)
f[p_, e_] := 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, x, sumdiv(n, y, sumdiv(n, z, sumdiv(n, t, eulerphi(x)*eulerphi(y)*eulerphi(z)*eulerphi(t)/eulerphi(lcm([x, y, z, t])))))); \\ Michel Marcus, Feb 26 2018

a160891(n) = sumdiv(n, d, moebius(n/d)*d^4)/eulerphi(n);
a(n) = sumdiv(n, d, a160891(d)); \\ Seiichi Manyama, May 12 2021

a(n) = Sum_{a|n, b|n, c|n, d|n} phi(a)*phi(b)*phi(c)*phi(d)/phi(lcm(a, b, c, d)), where phi is Euler totient function (cf. A000010).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)).
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.5010902655... . (End)
a(n) = Sum_{d divides n} J_4(d)/phi(d) = Sum_{1 <= i, j, k, l <= n} 1/phi(n/gcd(i,j,k,l,n)), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 23 2024

A280162 Number of subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 67, 212, 1983, 1120, 14204, 3652, 43339, 24033, 75040, 19156, 420396, 35872, 244684, 237440, 821335, 99472, 1610211, 152404, 2220960, 774224, 1283452, 318532, 9187868, 810969, 2403424, 2222704, 7241916, 783904, 15908480, 1016836, 14445411, 4061072, 6664624, 4090240, 47657439, 2031712

Offset: 1

Laszlo Toth, Dec 27 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems
G. A. Miller, On the subgroups of an abelian group, The Annals of Mathematics, 2nd Ser. 6:1 (1904), pp. 1-6.
L. Toth, The number of subgroups of the group Z_m x Z_n x Z_r x Z_s, arXiv:1611.03302 [math.GR], (2016).

Cf. A060724, A064803.

\\ For numsubgrp, see the Alekseyev link.
a(n)=my(f=factor(n)); prod(i=1,#f~, numsubgrp(f[i,1],f[i,2]*[1,1,1,1])) \\ Charles R Greathouse IV, Dec 27 2016

Terms a(32) and beyond from Charles R Greathouse IV, Dec 27 2016

A240547 Number of non-congruent solutions of x^2 + y^2 + z^2 + t^2 == 0 mod n.

Original entry on oeis.org

1, 8, 33, 32, 145, 264, 385, 128, 945, 1160, 1441, 1056, 2353, 3080, 4785, 512, 5185, 7560, 7201, 4640, 12705, 11528, 12673, 4224, 18625, 18824, 26001, 12320, 25201, 38280, 30721, 2048, 47553, 41480, 55825, 30240, 51985, 57608, 77649, 18560, 70561, 101640

Offset: 1

Laszlo Toth, Apr 07 2014

			For n=2 the a(2)=8 solutions are (0,0,0,0), (1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1), (1,1,1,1).

David A. Corneth, Table of n, a(n) for n = 1..10000
László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv preprint arXiv:1404.4214 [math.NT], 2014.
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.
Index to sequences related to sums of squares.

Cf. A020478, A069097, A086933, A087687, A208895, A229179.

A240547 := proc(n) local a, x, y, z, t ; a := 0 ; for x from 0 to n-1 do for y
from 0 to n-1 do for z from 0 to n-1 do for t from 0 to n-1 do if
(x^2+y^2+z^2+t^2) mod n = 0 mod n then a := a+1 ; fi; od; od ; od; od;
a ; end proc;
# alternative
A240547 := proc(n)
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := a*p^(2*e+1) ;
        else
            a := a* p^(2*e-1)*(p^(e+1)+p^e-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A240547(n),n=1..100) ; # R. J. Mathar, Jun 25 2018

b[2, e_] := 2^(2 e + 1);
b[p_, e_] := p^(2 e - 1)*(p^(e + 1) + p^e - 1);
a[n_] := Times @@ b @@@ FactorInteger[n];
Array[a, 42] (* Jean-François Alcover, Dec 05 2017 *)

a(n) = my(m); if( n<1, 0, forvec( v = vector(4, i, [0, n-1]), m += (0 == norml2(v)%n))); m /* Michael Somos, Apr 07 2014 */

a(n) = {my(f = factor(n), res = 1, start = 1, p, e, i); if(n % 2 == 0, res = 1<<(f[1,2]<<1+1); start = 2); for(i = start, #f~, p = f[i, 1]; e = f[i, 2]; res*=(p^(e<<1-1)*(p^(e+1)+p^e-1))); res} \\ David A. Corneth, Jul 22 2018

Multiplicative, with a(2^e) = 2^(2e+1) for e>=1, a(p^e) = p^(2e-1)*(p^(e+1)+p^e-1) for p > 2, e>=1.
For odd n, a(n) = A069097(n)*n = A020478(n). - R. J. Mathar, Jun 23 2018
Sum_{k=1..n} a(k) ~ c * n^4 + O(n^3 * log(n)), where c = 5*Pi^2/(168*zeta(3)) = 0.244362... (Tóth, 2014). - Amiram Eldar, Oct 18 2022

A200758 Superimperfect numbers.

Original entry on oeis.org

2, 4, 8, 128, 32768, 2147483648

Offset: 1

Laszlo Toth, Nov 22 2011

A number n is said to be superimperfect if 2*beta(beta(n)) = n, where beta is the multiplicative function defined by beta(p^e) = p^e - p^(e-1) + p^(e-2) - ... + (-1)^e for every prime power p^e. The function beta is called the alternating sum-of-divisors function. Here beta(n) is the absolute value of A061020(n). There are no other superimperfect numbers up to 10^7. The number 2^(2^k-1) is superimperfect if and only if k=1,2,3,4,5.

Laszlo Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
Laszlo Toth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siofok, Hungary.

Cf. A061020, A127724, A127725, A019279, A058891, A206369.

beta(n)=sumdiv(n,d,(-1)^bigomega(n/d)*d)
for(n=1,1e8,if(2*beta(beta(n))==n,print1(n", "))) \\ Charles R Greathouse IV, Nov 22 2011

ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
beta(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2]))
is(n)=my(b=beta(n)); 2*b-2 >= n && 2*beta(b)==n \\ Charles R Greathouse IV, Dec 27 2016

A199806 Alternating LCM-sum: a(n) = Sum_{k=1..n} (-1)^(k-1)*lcm(k,n).

Original entry on oeis.org

1, 0, 0, 8, -5, 18, -14, 80, -9, 100, -44, 204, -65, 294, 30, 672, -119, 540, -152, 1040, 63, 1210, -230, 1752, -75, 2028, -54, 2996, -377, 2190, -434, 5440, 165, 4624, 280, 5472, -629, 6498, 234, 8800, -779, 6300, -860, 12188, 225, 11638, -1034, 14256, -245, 13000

Offset: 1

Laszlo Toth, Nov 10 2011

sign

Amiram Eldar, Table of n, a(n) for n = 1..10000
Laszlo Toth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7

Cf. A051193, A199084.

[&+[(-1)^(k-1)*Lcm(k,n):k in [1..n]]: n in [1..50]]; // Marius A. Burtea, Oct 02 2019

Table[Sum[(-1)^(k-1) LCM[k,n],{k,n}],{n,50}] (* Harvey P. Dale, Jan 30 2024 *)

a(n)=-sum(k=1,n,(-1)^k*lcm(k,n)) \\ Charles R Greathouse IV, Nov 10 2011

def A199806(n) : return add((-1)^(k-1)*lcm(k,n) for k in (1..n))
[A199806(n) for n in (1..50)]  # Peter Luschny, Nov 10 2011

A166234 The inverse of the constant 1 function under the exponential convolution (also called the exponential Möbius function).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 0, 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, 0, -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, 0, 0, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1

Offset: 1

Laszlo Toth, Oct 09 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Xiaodong Cao and Wenguang Zahi, Some arithmetic functions involving exponential divisors, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.7.
Andrew V. Lelechenko, Exponential and infinitary divisors, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; arXiv preprint, arXiv:1405.7597 [math.NT], 2014, function mu^(E)(n).
M. V. Subbarao, On some arithmetic convolutions, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions, Lecture Notes in Mathematics No. 251, Springer, 1972, pp. 247-271; alternative link.
László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-296; arXiv preprint, arXiv:math/0610274 [math.NT], 2006-2009.
László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Cf. A008683, A049419, A051377, A124010, A209802 (partial sums).

a166234 = product . map (a008683 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

A166234 := proc(n)
    local a,p;
    a := 1;
    if n =1 then
        ;
    else
        for p in ifactors(n)[2] do
                    a := a*numtheory[mobius](op(2,p)) ;
        end do:
    end if;
    a ;
end proc:# R. J. Mathar, Nov 30 2016

a[n_] := Times @@ MoebiusMu /@ FactorInteger[n][[All, 2]];
Array[a, 100] (* Jean-François Alcover, Nov 16 2017 *)

a(n)=factorback(apply(moebius, factor(n)[,2])) \\ Charles R Greathouse IV, Sep 02 2015

Multiplicative, a(p^e) = mu(e) for any prime power p^e (e>=1), where mu is the Möbius function A008683.
a(A130897(n)) = 0; a(A209061(n)) <> 0. - Reinhard Zumkeller, Mar 13 2012
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Product_{p prime} (1 + Sum_{k>=2} (mu(k) - mu(k-1))/p^k) = 0.3609447238... (Tóth, 2007). - Amiram Eldar, Nov 08 2020

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.

Original entry on oeis.org

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

A143869 An integer k is called regular (mod n) if there is an integer x such that k^2 x == k (mod n). Then these numbers are the sum of regular integers k (mod n) such that 1 <= k <= n for n=1,2,... .

Original entry on oeis.org

1, 3, 6, 8, 15, 21, 28, 24, 36, 55, 66, 60, 91, 105, 120, 80, 153, 135, 190, 160, 231, 253, 276, 192, 275, 351, 270, 308, 435, 465, 496, 288, 561, 595, 630, 396, 703, 741, 780, 520, 861, 903, 946, 748, 810, 1081, 1128, 672, 1078, 1075, 1326, 1040, 1431, 1053

Offset: 1

Laszlo Toth, Sep 04 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Osama Alkam and Emad Abu Osba, On the regular elements in Zn, Turk J Math, 32 (2008), 31-39.
B. Apostol and L. Petrescu, Extremal Orders of Certain Functions Associated with Regular Integers (mod n), Journal of Integer Sequences, 2013, # 13.7.5.
L. Tóth, Regular integers modulo n, Annales Univ. Sci. Budapest., Sect. Comp., 29 (2008), 263-275.

Cf. A055653, A176345.

isregu(k, n) = {g = gcd(k, n); if ((n % g == 0) && (gcd(g, n/g) == 1), return(k), return(0));}
a(n) = sum(k=1, n, isregu(k, n)) \\ Michel Marcus, May 25 2013

a(n) = n*(A055653(n)+1)/2.

Extended by R. J. Mathar, Sep 05 2008

A130897 Numbers that are not exponentially squarefree.

Original entry on oeis.org

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 256, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 768, 784, 810, 816, 848, 880, 891, 912, 944, 976

Offset: 1

Laszlo Toth, Mar 18 2011

nonn

A positive integer is called exponentially squarefree (e-squarefree) if in its prime power factorization all the exponents are squarefree.
a(n) is the sequence of positive integers in which prime power factorization there is at least one nonsquarefree exponent.
n is non-e-squarefree iff f(n)=0, where f(n) is the exponential Moebius function A166234.
Product_{k = 1..A001221(n)} A008966(A124010(n,k)) = 0. - Reinhard Zumkeller, Mar 13 2012
The density of {a(n)} is 0.04407699... (see comment in A209061). - Peter J. C. Moses and Vladimir Shevelev, Sep 08 2015

			16=2^4, 48=2^4*3, 256=2^8 are non-e-squarefree, since 4 and 8 are nonsquarefree.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. V. Subbarao, On some arithmetic convolutions, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics No. 251, 247-271, Springer, 1972, doi:10.1007/BFb0058796.
Laszlo Toth, On certain arithmetic functions involving exponential divisors, II., Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166.

Complement of A209061; subsequence of A013929, A046099, and A046101.

Cf. A005117, A166234, A049419, A051377, A008966, A124010.

a130897 n = a130897_list !! (n-1)
a130897_list = filter
   (any (== 0) . map (a008966 . fromIntegral) . a124010_row) [1..]
-- Reinhard Zumkeller, Mar 13 2012

filter:=n ->  not andmap(t -> numtheory:-issqrfree(t[2]), ifactors(n)[2]);
select(filter, [$1..1000]); # Robert Israel, Sep 03 2015

Select[Range@ 1000, ! AllTrue[Last /@ FactorInteger@ #, SquareFreeQ] &] (* Michael De Vlieger, Sep 07 2015, Version 10 *)

is(n)=my(f=factor(n)[, 2]); for(i=1, #f, if(!issquarefree(f[i]), return(1))); 0 \\ Charles R Greathouse IV, Sep 03 2015

cp's OEIS Frontend

User: Laszlo Toth

A280184 Number of cyclic subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

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A280162 Number of subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

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A240547 Number of non-congruent solutions of x^2 + y^2 + z^2 + t^2 == 0 mod n.

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A200758 Superimperfect numbers.

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A199806 Alternating LCM-sum: a(n) = Sum_{k=1..n} (-1)^(k-1)*lcm(k,n).

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A166234 The inverse of the constant 1 function under the exponential convolution (also called the exponential Möbius function).

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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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