A053819 -id:A053819 - 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

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

A308477 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n, gcd(n,j) = 1} j^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 4, 1, 1, 9, 10, 10, 2, 1, 1, 17, 28, 30, 6, 6, 1, 1, 33, 82, 100, 26, 21, 4, 1, 1, 65, 244, 354, 126, 91, 16, 6, 1, 1, 129, 730, 1300, 626, 441, 84, 27, 4, 1, 1, 257, 2188, 4890, 3126, 2275, 496, 159, 20, 10, 1, 1, 513, 6562, 18700, 15626, 12201, 3108, 1053, 140, 55, 4

Offset: 1

Ilya Gutkovskiy, May 29 2019

			Square array begins:
  1,   1,   1,    1,    1,     1,  ...
  1,   1,   1,    1,    1,     1,  ...
  2,   3,   5,    9,   17,    33,  ...
  2,   4,  10,   28,   82,   244,  ...
  4,  10,  30,  100,  354,  1300,  ...
  2,   6,  26,  126,  626,  3126,  ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..4 give A000010, A023896, A053818, A053819, A053820.

Cf. A103438.

Table[Function[k, Sum[If[GCD[n, j] == 1, j^k, 0], {j, 1, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

A295574 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^2.

Original entry on oeis.org

0, 1, 1, 1, 5, 1, 14, 10, 21, 10, 55, 26, 91, 35, 70, 84, 204, 75, 285, 140, 210, 165, 506, 196, 525, 286, 549, 406, 1015, 340, 1240, 680, 880, 680, 1190, 654, 2109, 969, 1482, 1080, 2870, 966, 3311, 1650, 2010, 1771, 4324, 1544, 4214, 2050

Offset: 1

N. J. A. Sloane, Dec 08 2017

nonn

n does not divide a(n) iff n = (2^k)*(q^m) with k > 0, m >= 0 and q odd prime such that q == 3 (mod 4) or n = (2^k)*(3^L)*Product_{q} q^(v_q) with k >= 0, L > 0, v_q >= 0 and all q odd primes such that q == 5 (mod 6). - René Gy, Oct 21 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.
René Gy, The sum of product pairs of integers prime to n, Math StackExchange.

In the Baum (1982) paper, S_1, S_2, S_3, S_4 are A023896, A053818, A053819, A053820, and S'_1, S'_2, S'_3, S'_4 are A066840, A295574, A295575, A295576.

Cf. A023022.

R:=proc(n,k) local x,t1,S;
t1:={}; S:=0;
for x from 1 to floor(n/2) do if gcd(x,n)=1 then t1:={op(t1),x^k}; S:=S+x^k; fi; od;
S; end;
s:=k->[seq(R(n,k),n=1..50)];
s(2);

f[n_] := Plus @@ (Select[ Range[n/2], GCD[#, n] == 1 &]^2); Array[f, 50] (* Robert G. Wilson v, Dec 10 2017 *)

a(n) = sum(j=1, n\2, (gcd(j, n)==1)*j^2); \\ Michel Marcus, Dec 10 2017

A295575 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^3.

Original entry on oeis.org

0, 1, 1, 1, 9, 1, 36, 28, 73, 28, 225, 126, 441, 153, 416, 496, 1296, 469, 2025, 1100, 1710, 1225, 4356, 1800, 4959, 2556, 5581, 4410, 11025, 3872, 14400, 8128, 11090, 8128, 15822, 8910, 29241, 13041, 21996, 16400, 44100, 15426, 53361, 27830, 33716, 29161, 76176, 27936, 77652, 37828

Offset: 1

N. J. A. Sloane, Dec 08 2017

nonn

If p is an odd prime, a(p) = (n^2-1)^2/64. - Robert Israel, Dec 10 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.

In the Baum (1982) paper, S_1, S_2, S_3, S_4 are A023896, A053818, A053819, A053820, and S'_1, S'_2, S'_3, S'_4 are A066840, A295574, A295575, A295576.

f:= n -> add(t^3, t = select(t->igcd(t,n)=1, [$1..n/2])) :
map(f, [$1..100]); # Robert Israel, Dec 10 2017

f[n_] := Plus @@ (Select[Range[n/2], GCD[#, n] == 1 &]^3); Array[f, 50] (* Robert G. Wilson v, Dec 10 2017 *)

a(n) = sum(j=1, n\2, (gcd(j, n)==1)*j^3); \\ Michel Marcus, Dec 10 2017

A295576 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^4.

Original entry on oeis.org

0, 1, 1, 1, 17, 1, 98, 82, 273, 82, 979, 626, 2275, 707, 2674, 3108, 8772, 3027, 15333, 9044, 14994, 9669, 39974, 17668, 50085, 24310, 60597, 50470, 127687, 45604, 178312, 103496, 149908, 103496, 225302, 129750, 432345, 187017, 349830, 266088, 722666

Offset: 1

N. J. A. Sloane, Dec 08 2017

nonn

If p is an odd prime, a(p) = p*(p^2-1)*(3*p^2-7)/480. - Robert Israel, Dec 10 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.

In the Baum (1982) paper, S_1, S_2, S_3, S_4 are A023896, A053818, A053819, A053820, and S'_1, S'_2, S'_3, S'_4 are A066840, A295574, A295575, A295576.

f:= n -> add(t^4, t = select(t->igcd(t,n)=1, [$1..n/2])):
map(f, [$1..100]); # Robert Israel, Dec 10 2017

f[n_] := Plus @@ (Select[Range[n/2], GCD[#, n] == 1 &]^4); Array[f, 41] (* Robert G. Wilson v, Dec 10 2017 *)

a(n) = sum(j=1, n\2, (gcd(j, n)==1)*j^4); \\ Michel Marcus, Dec 10 2017

A343513 a(n) = Sum_{k=1..n} (k/gcd(n, k))^3.

Original entry on oeis.org

1, 2, 10, 30, 101, 137, 442, 526, 1063, 1202, 3026, 1965, 6085, 4853, 7310, 8654, 18497, 10100, 29242, 17630, 29557, 30857, 64010, 30397, 77601, 60842, 89272, 71913, 164837, 60737, 216226, 139470, 188165, 180338, 265142, 152544, 443557, 282665, 371134, 275726, 672401, 251066, 815410, 461645

Offset: 1

Ilya Gutkovskiy, Apr 17 2021

nonn

a(n) = 1+n^2*(n-1)^2/4 if n is prime. - Robert Israel, Apr 19 2021

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A053819, A057661, A332654, A343497, A343514.

f:= proc(n) local k;
  add((k/igcd(n,k))^3,k=1..n)
end proc:
map(f, [$1..100]); # Robert Israel, Apr 19 2021

Table[Sum[(k/GCD[n, k])^3, {k, 1, n}], {n, 1, 44}]

a(n) = sum(k=1, n, (k/gcd(n, k))^3); \\ Michel Marcus, Apr 17 2021

a(n) = Sum_{d|n} A053819(d).

A057789 a(n) = Sum_{k = 1..n, gcd(k,n)=1} k*(n-k).

Original entry on oeis.org

0, 1, 4, 6, 20, 10, 56, 44, 84, 60, 220, 92, 364, 182, 280, 344, 816, 318, 1140, 520, 840, 770, 2024, 760, 2100, 1300, 2196, 1540, 4060, 1240, 4960, 2736, 3520, 2992, 4760, 2580, 8436, 4218, 5928, 4240, 11480, 3612, 13244, 6380, 8040, 7590, 17296, 6128

Offset: 1

Leroy Quet, Nov 04 2000

Equal to convolution sum over positive integers, k, where k<=n and gcd(k,n)=1, except in first term, where the convolution sum is 1 instead of 0.

			Since 1, 3, 5 and 7 are relatively prime to 8 and are <= 8, a(8) = 1*(8-1) +3*(8-3) +5*(8-5) +7*(8-7) = 44.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000292, A023896, A053818, A053819, A002618, A053191, A023900.

f:= proc(n) local i;
  2*add(`if`(igcd(i,n)=1, i*(n-i),0),i=1..n/2)
end proc:
f(2):= 1:
map(f, [$1..100]); # Robert Israel, Sep 29 2019

a[n_] := 2 Sum[Boole[CoprimeQ[k, n]] k (n - k), {k, 1, n/2}];
a[2] = 1;
Array[a, 100] (* Jean-François Alcover, Aug 16 2020, after Maple *)

a(n) = sum(k=1, n, if (gcd(n,k)==1, k*(n-k))); \\ Michel Marcus, Sep 29 2019

From Robert Israel, Sep 29 2019: (Start)
If n is prime, a(n) = A000292(n-1).
If n/2 is an odd prime, a(n) =  A000292(n-2)/2.
If n/3 is a prime other than 3, a(n) = A000292(n-3)*2*n/(3*(n-2)). (End)
From Ridouane Oudra, Mar 21 2024: (Start)
a(n) = n*A023896(n) - A053818(n) ;
a(n) = (2/3)*(n*A023896(n) - A053819(n)/n) ;
a(n) = (n/6)*(A002618(n) - A023900(n)) ;
a(n) = (1/6)*(A053191(n) - n*A023900(n)). (End)
Sum_{k=1..n} a(k) ~ n^4 / (4*Pi^2). - Amiram Eldar, Apr 11 2024

A057792 Sum[k^k], where sum is over positive integers, k, where k <= n and gcd(k,n) = 1.

Original entry on oeis.org

1, 1, 5, 28, 288, 3126, 50069, 826696, 17604145, 388244060, 10405071317, 285312497280, 9211817190184, 303160805686506, 11415167261421900, 438197051187369424, 18896062057839751444, 827240565046755853710

Offset: 1

Leroy Quet, Nov 04 2000

nonn

			a(4) = 1^1 + 3^3 = 28, since 1 and 3 are the positive integers <= 4 and relatively prime to 4.

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A023896, A053818, A053819, A053820, A308481.

a(n) = sum(k=1, n, if (gcd(k, n)==1, k^k)); \\ Michel Marcus, Jun 19 2021

cp's OEIS Frontend

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

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

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A308477 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n, gcd(n,j) = 1} j^k.

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A295574 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^2.

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A295575 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^3.

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A295576 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^4.

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A343513 a(n) = Sum_{k=1..n} (k/gcd(n, k))^3.

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