A308477 -id:A308477 - cp's OEIS frontend

A053818 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^2.

1, 1, 5, 10, 30, 26, 91, 84, 159, 140, 385, 196, 650, 406, 620, 680, 1496, 654, 2109, 1080, 1806, 1650, 3795, 1544, 4150, 2756, 4365, 3164, 7714, 2360, 9455, 5456, 7370, 6256, 9940, 5196, 16206, 8778, 12324, 8560, 22140, 6972, 25585

Offset: 1

N. J. A. Sloane, Apr 07 2000

nonn

Equals row sums of triangle A143612. - Gary W. Adamson, Aug 27 2008
a(n) = A175505(n) * A023896(n) / A175506(n). For number n >= 1 holds B(n) = a(n) / A023896(n) = A175505(n) / A175506(n), where B(n) = antiharmonic mean of numbers k such that GCD(k, n) = 1 for k < n. - Jaroslav Krizek, Aug 01 2010
n does not divide a(n) iff n is a term in A316860, that is, either n is a power of 2 or n is a multiple of 3 and no prime factor of n is congruent to 1 mod 3. - Jianing Song, Jul 16 2018

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_2(n).
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #2.

G. C. Greubel, Table of n, a(n) for n = 1..2500
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
Geoffrey B. Campbell, Dirichlet summations and products over primes, Int. J. Math. Math. Sci. 16 92) (1993) 359. eq. (3.1)
Constantin M. Petridi, The Sums of the k-powers of the Euler set and their connection with Artin's conjecture for primitive roots, arXiv:1612.07632 [math.NT], 2016-2018.

Cf. A023896, A053819, A053820.
Cf. A143612. - Gary W. Adamson, Aug 27 2008
Cf. A179871-A179880, A179882-A179887, A179890, A179891, A007645, A003627, A034934. - Jaroslav Krizek, Aug 01 2010
Column k=2 of A308477.

A053818 := proc(n)
    local a,k;
    a := 0 ;
    for k from 1 to n do
        if igcd(k,n) = 1 then
            a := a+k^2 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 26 2013

a[n_] := Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2); Array[a, 43] (* Robert G. Wilson v, Jul 01 2010 *)
a[1] = 1; a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (n^2/3) * Times @@ ((p - 1)*p^(e - 1)) + (n/6) * Times @@ (1 - p)]; Array[a, 100] (* Amiram Eldar, Dec 03 2023 *)

a(n) = sum(k=1, n, k^2*(gcd(n,k) == 1)); \\ Michel Marcus, Jan 30 2016

a(n) = {my(f = factor(n)); if(n == 1, 1, (n^2/3) * eulerphi(f) + (n/6) * prod(i = 1, #f~, 1 - f[i, 1]));} \\ Amiram Eldar, Dec 03 2023

If n = p_1^e_1 * ... *p_r^e_r then a(n) = n^2*phi(n)/3 + (-1)^r*p_1*..._p_r*phi(n)/6.
a(n) = n^2*A000010(n)/3 + n*A023900(n)/6, n>1. [Brown]
a(n) = (A000010(n)/3) * (n^2 + (-1)^A001221(n)*A007947(n)/2) for n>=2. - Jaroslav Krizek, Aug 24 2010
G.f. A(x) satisfies: A(x) = x*(1 + x)/(1 - x)^4 - Sum_{k>=2} k^2 * A(x^k). - Ilya Gutkovskiy, Mar 29 2020
Sum_{k=1..n} a(k) ~ n^4 / (2*Pi^2). - Amiram Eldar, Dec 03 2023

A053819 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^3.

Original entry on oeis.org

1, 1, 9, 28, 100, 126, 441, 496, 1053, 1100, 3025, 1800, 6084, 4410, 7200, 8128, 18496, 8910, 29241, 16400, 29106, 27830, 64009, 27936, 77500, 54756, 88209, 67032, 164836, 52200, 216225, 130816, 185130, 161840, 264600, 140616, 443556

Offset: 1

N. J. A. Sloane, Apr 07 2000

nonn

Except for a(2) = 1, a(n) is always divisible by n. - Jianing Song, Jul 13 2018

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_3(n).

Robert Israel, Table of n, a(n) for n = 1..10000
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (2005) 403-408.
Geoffrey B. Campbell, Dirichlet summations and products over primes, Int. J. Math. Math. Sci. 16 92) (1993) 359. eq. (3.1)
Constantin M. Petridi, The Sums of the k-powers of the Euler set and their connection with Artin's conjecture for primitive roots, arXiv:1612.07632 [math.NT], 2016-2018.

Cf. A000010, A001221, A007947, A023900, A053818, A053820.

Column k=3 of A308477.

f:= proc(n) local F,t;
  F:= ifactors(n)[2];
  numtheory:-phi(n)*(n^3 + (-1)^nops(F)*mul(t[1],t=F)*n)/4
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Jan 29 2018

Table[Sum[j^3, {j, Select[Range[n], GCD[n, #] == 1 &]}], {n, 1, 37}] (* Geoffrey Critzer, Mar 03 2015 *)
a[1] = 1; a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (n^2/4) * (n * Times @@ ((p - 1)*p^(e - 1)) + Times @@ (1 - p))]; Array[a, 100] (* Amiram Eldar, Dec 03 2023 *)

a(n) = sum(k=1,n, k^3*(gcd(n,k)==1)); \\ Michel Marcus, Mar 03 2015

a(n) = {my(f = factor(n)); if(n == 1, 1, (n^2/4) * (n * eulerphi(f) + prod(i = 1, #f~, 1 - f[i, 1])));} \\ Amiram Eldar, Dec 03 2023

a(n) = n^2/4*(n*A000010(n) + A023900(n)), n > 1. - Vladeta Jovovic, Apr 17 2002
a(n) = eulerphi(n)*(n^3 + (-1)^omega(n)*rad(n)*n)/4. See Petridi link. - Michel Marcus, Jan 29 2017
G.f. A(x) satisfies: A(x) = x*(1 + 4*x + x^2)/(1 - x)^5 - Sum_{k>=2} k^3 * A(x^k). - Ilya Gutkovskiy, Mar 29 2020
Sum_{k=1..n} a(k) ~ 3 * n^5 / (10*Pi^2). - Amiram Eldar, Dec 03 2023

A053820 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^4.

Original entry on oeis.org

1, 1, 17, 82, 354, 626, 2275, 3108, 7395, 9044, 25333, 17668, 60710, 50470, 88388, 103496, 243848, 129750, 432345, 266088, 497574, 497178, 1151403, 539912, 1541770, 1153724, 1900089, 1516844, 3756718, 1246568, 5273999

Offset: 1

N. J. A. Sloane, Apr 07 2000

nonn

If gcd(n,30) = 1, then a(n) is divisible by n. If n has at least one prime factor == 1 (mod 30), then a(n) is divisible by n. - Jianing Song, Jul 13 2018

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_4(n).
L. E. Dickson, History of the Theory of Numbers, Vol. I (Reprint 1966), p. 140.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (2005) 403-408.

Column k=4 of A308477.

Cf. A000010, A023900, A053818, A053819, A063453.

a[n_] := Sum[If[GCD[n, k] == 1, k^4, 0], {k, 1, n}]; Table[a[n], {n, 1, 31}] (* Jean-François Alcover, Feb 26 2014 *)
a[1] = 1; a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (n^4/5) * Times @@ ((p - 1)*p^(e - 1)) + (n^3/3) * Times @@ (1 - p) - (n/30) * Times @@ (1 - p^3)]; Array[a, 100] (* Amiram Eldar, Dec 03 2023 *)

a(n) = sum(k=1, n, (gcd(n,k) == 1)*k^4); \\ Michel Marcus, Feb 26 2014

a(n) = {my(f = factor(n)); if(n == 1, 1, (n^4/5) * eulerphi(f) + (n^3/3) * prod(i = 1, #f~, 1 - f[i, 1]) - (n/30) * prod(i = 1, #f~, 1 - f[i, 1]^3));} \\ Amiram Eldar, Dec 03 2023

a(n) = (6*n^4*A000010(n)+10*n^3*A023900(n)-n*A063453(n))/30 for n>1. Formula is derived from a more general formula of A. Thacker (1850), see [Dickson, Brown]. - Franz Vrabec, Aug 21 2005
G.f. A(x) satisfies: A(x) = x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^6 - Sum_{k>=2} k^4 * A(x^k). - Ilya Gutkovskiy, Mar 29 2020
Sum_{k=1..n} a(k) ~ n^6 / (5*Pi^2). - Amiram Eldar, Dec 03 2023

A308481 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^n.

Original entry on oeis.org

1, 1, 9, 82, 1300, 15626, 376761, 6161988, 176787117, 3769318700, 142364319625, 3152513804548, 154718778284148, 4340009120036086, 210971169748692000, 7281661100510001416, 435659030617933827136, 14181101408561469791694, 1052864393300587929716721, 41673894815421072916530408

Offset: 1

Ilya Gutkovskiy, May 30 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

First superdiagonal of A308477.

Cf. A031971.

a[n_] := Sum[If[GCD[n, k] == 1, k^n, 0], {k, 1, n}]; Table[a[n], {n, 1, 20}]

a(n) = sum(k=1, n, (gcd(n,k)==1)*k^n) \\ Felix Fröhlich, May 30 2019

cp's OEIS Frontend

A053818 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^2.

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A053819 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^3.

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A053820 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^4.

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A308481 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^n.

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