A057660 -id:A057660 - cp's OEIS frontend

A174405 Partial sums of Sum_{k=1..n} n/gcd(n,k), or partial sums of Sum_{d|n} d*phi(d) (see A057660).

1, 4, 11, 22, 43, 64, 107, 150, 211, 274, 385, 462, 619, 748, 895, 1066, 1339, 1522, 1865, 2096, 2397, 2730, 3237, 3538, 4059, 4530, 5077, 5550, 6363, 6804, 7735, 8418, 9195, 10014, 10917, 11588, 12921, 13950, 15049, 15952, 17593, 18496, 20303, 21524, 22805, 24326, 26489, 27686, 29787, 31350, 33261, 34988

Offset: 1

Jonathan Vos Post, Nov 27 2010

nonn

The subsequence of primes in this sequence begins 11, 43, 107, 211, 619, 5077, 26489. The subsequence of squares in this sequence begins 1, 4, 64, 18496 = 2^6 * 17^2.

			a(9) = 1 + 3 + 7 + 11 + 21 + 21 + 43 + 43 + 61 = 211 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Marko Riedel, answer to 'Euler phi function, number theory', MathStackExchange, 2014.
Günter Rote, Moritz Rüber, and Morteza Saghafian, Grid Peeling of Parabolas, arXiv:2402.15787 [cs.CG], 2024. See p. 9.
J. Sándor and A. V. Kramer, Über eine zahlentheoretische Funktion. Mathematica Moravica, 3 (1999), 53-62.

Cf. A000010, A057660.

f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Accumulate @ Array[a, 100] (* Amiram Eldar, Nov 21 2020 *)

a(n)=sum(k=1,n,sumdiv(k,d,eulerphi(d)*d)) \\ Charles R Greathouse IV, May 21 2014

a(n) = sum(k=1, n, k * eulerphi(k) * (n\k)); \\ Michel Marcus, May 30 2018

a(n) = Sum_{i=1..n} A057660(i) = Sum_{i=1..n} Sum_{k=1..i} i/gcd(i,k) = Sum_{i=1..n} ( Sum_{d|i} A000010(d^2) ) = Sum_{i=1..n} ( Sum_{d|i} d*A000010(d) ) = Sum_{i=1..n} (sum of the orders of the elements in a cyclic group with i elements).
Riedel, and Sándor&Kramer give a(n) ~ 2/Pi^2 * zeta(3) * n^3. - Charles R Greathouse IV, May 21 2014; Günter Rote, Nov 03 2021
G.f.: (1/(1 - x))*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, Aug 31 2017
a(n) = Sum_{k=1..n} k * phi(k) * floor(n/k), where phi(k) is the Euler totient function. - Daniel Suteu, May 30 2018

A308471 Lowest outliers for A057660.

Original entry on oeis.org

1, 4, 6, 12, 24, 30, 60, 120, 180, 210, 360, 420, 840, 1260, 2520, 4620, 9240, 13860, 27720, 55440, 60060, 120120, 180180, 360360, 720720, 1441440, 1801800, 2042040, 3063060, 6126120, 12252240, 24504480, 30630600, 36756720, 38798760

Offset: 1

Charlie Neder, May 29 2019

nonn

A057660(n) is a multiplicative function bounded above by n*(n-1)+1, which is reached whenever n is 1 or prime. These numbers are the n such that the ratio between A057660(n) and the upper bound reaches a record low.
Motivated by Daniel Forgues's conjecture that this sequence consists of 4 and A051451.
A subsequence of A025487.

			A057660(60060)/(60060*60059+1) = 1211716737/3607143541 ~ 0.3359214, and every number less than 60060 has a ratio > 0.34, so 60060 is in this sequence.

Charlie Neder, Table of n, a(n) for n = 1..215

Cf. A057660, A025487, A051451.

A318444 Numerators of the sequence whose Dirichlet convolution with itself yields A057660(n) = Sum_{k=1..n} n/gcd(n,k).

Original entry on oeis.org

1, 3, 7, 35, 21, 21, 43, 239, 195, 63, 111, 245, 157, 129, 147, 6851, 273, 585, 343, 735, 301, 333, 507, 1673, 1643, 471, 3011, 1505, 813, 441, 931, 50141, 777, 819, 903, 6825, 1333, 1029, 1099, 5019, 1641, 903, 1807, 3885, 4095, 1521, 2163, 47957, 6555, 4929, 1911, 5495, 2757, 9033, 2331, 10277, 2401, 2439, 3423

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Because A057660 contains only odd values, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.

General formula: if k >= 0, m > 0, and the Dirichlet generating function is zeta(s-k)^m * f(s), where f(s) has all possible poles at points less than k+1, then Sum_{j=1..n} a(j) ~ n^(k+1) * log(n)^(m-1) * f(k+1) / ((k+1) * Gamma(m)) * (1 + (m-1)*(m*gamma - 1/(k+1) + f'(k+1)/f(k+1)) / log(n)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function. - Vaclav Kotesovec, May 10 2025

Antti Karttunen, Table of n, a(n) for n = 1..16384
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A057660, A046644 (denominators).

Cf. also A318443.

a57660[n_] := DivisorSigma[2, n^2]/DivisorSigma[1, n^2];
f[1] = 1; f[n_] := f[n] = 1/2 (a57660[n] - Sum[f[d]*f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
Table[f[n] // Numerator, {n, 1, 60}] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 16384;
A057660(n) = sumdivmult(n, d, eulerphi(d)*d); \\ From A057660
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318444aux = DirSqrt(vector(up_to, n, A057660(n)));
A318444(n) = numerator(v318444aux[n]);

for(n=1, 100, print1(numerator(direuler(p=2, n, ((1-p*X)/((1-p^2*X)*(1-X)))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A057660(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318444(k) / A046644(k) ~ n^3 * Pi^(-3/2) * sqrt(2*zeta(3)/(3*log(n))) * (1 + (1/3 - gamma/2 + 3*zeta'(2)/Pi^2 - zeta'(3)/(2*zeta(3))) / (2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 10 2025

A377585 E.g.f.: exp(Sum_{k>=1} A057660(k) * x^k).

Original entry on oeis.org

1, 1, 7, 61, 577, 7381, 96511, 1619857, 28368481, 560654857, 12100090231, 282510616741, 7098784113697, 190647458125021, 5461212525476527, 165494332157561401, 5306572876379307841, 178898083900878623377, 6336492991778941139431, 234867483921621706900237, 9096385945218131126509441

Offset: 0

Vaclav Kotesovec, Nov 02 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A057660.

nmax = 25; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k^2]/DivisorSigma[1, k^2]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ 3^(1/4) * zeta(3)^(1/8) * exp(sqrt(Pi)*n^(1/4)/(6^(3/2)*zeta(3)^(1/4)) + 2^(5/2)*zeta(3)^(1/4)*n^(3/4)/sqrt(3*Pi) - n) * n^(n - 1/8) / (2^(3/4) * Pi^(1/4)).

A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2p_i + 3p_i^2 + ... + (e_i+1)*p_i^e_i).

Original entry on oeis.org

1, 5, 7, 17, 11, 35, 15, 49, 34, 55, 23, 119, 27, 75, 77, 129, 35, 170, 39, 187, 105, 115, 47, 343, 86, 135, 142, 255, 59, 385, 63, 321, 161, 175, 165, 578, 75, 195, 189, 539, 83, 525, 87, 391, 374, 235, 95, 903, 162, 430, 245, 459, 107, 710, 253, 735, 273, 295, 119

Offset: 1

N. J. A. Sloane, Apr 17 2001

Equals row sums of triangle A143313. - Gary W. Adamson, Aug 06 2008
Equals row sums of triangle A127099. - Gary W. Adamson, Jul 27 2008
Sum of the divisors d2 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

			a(4) = a(2^2) = 1 + (2)*(2) + (3)*(2^2) = 17;
a(6) = a(2)*a(3) = (1 + (2)*(2))*(1+(2)*(3)) = (5)*(7) = 35.
a(6) = tau(1) + 2*tau(2) + 3*tau(3) + 6*tau(6) = 1 + 2*2 + 3*2 + 6*4 = 35.

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A000203, A001001, A006171, A038040 (Mobius transform), A049060, A057660, A057723, A327960 (Dirichlet inverse).

Cf. also triangles A027750, A127099, A143313.

a060640 n = sum [d * a000005 d | d <- a027750_row n]
-- Reinhard Zumkeller, Feb 29 2012

A060640 := proc(n) local ans, i, j; ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(1+sum((j+1)*ifactors(n)[2][i][1]^j,j=1..ifactors(n)[2][i][2])): od: RETURN(ans) end:

a[n_] := Total[#*DivisorSigma[1, n/#] & /@ Divisors[n]];
a /@ Range[59] (* Jean-François Alcover, May 19 2011, after Vladeta Jovovic *)
f[p_, e_] := ((e + 1)*p^(e + 2) - (e + 2)*p^(e + 1) + 1)/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 10 2022 *)

j=[]; for(n=1,200,j=concat(j,sumdiv(n,d,n/d*sigma(d)))); j

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p*X)^2)[n]) /* Ralf Stephan */

N=66; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j); t=1/prod(j=1,N, eta(x^(j)));
t=log(t);t=serconvol(t,c);
Vec(t) /* Joerg Arndt, May 03 2008 */

{ for (n=1, 1000, write("b060640.txt", n, " ", direuler(p=2, n, 1/(1 - X)/(1 - p*X)^2)[n]); ) } /* Harry J. Smith, Jul 08 2009 */

def A060640(n) :
    sigma = sloane.A000203
    return add(sigma(k)*(n/k) for k in divisors(n))
[A060640(i) for i in (1..59)] # Peter Luschny, Sep 15 2012

a(n) = Sum_{d|n} d*tau(d), where tau(d) is the number of divisors of d, cf. A000005. a(n) = Sum_{d|n} d*sigma(n/d), where sigma(n)=sum of divisors of n, cf. A000203. - Vladeta Jovovic, Apr 23 2001
Multiplicative with a(p^e) = ((e+1)*p^{e+2} - (e+2)*p^{e+1} + 1) / (p-1)^2. Dirichlet g.f.: zeta(s)*zeta(s-1)^2. - Franklin T. Adams-Watters, Aug 03 2006
L.g.f.: Sum(A060640(n)*x^n/n) = -log( Product_{j>=1} P(x^j) ) where P(x) = Product_{k>=1} (1-x^k). - Joerg Arndt, May 03 2008
G.f.: Sum_{k>=1} k*tau(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 06 2018
Sum_{k=1..n} a(k) ~ n^2/24 * ((4*gamma - 1)*Pi^2 + 2*Pi^2 * log(n) + 12*Zeta'(2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 01 2019

More terms from James Sellers, Vladeta Jovovic and Matthew Conroy, Apr 17 2001

A057661 a(n) = Sum_{k=1..n} lcm(n,k)/n.

Original entry on oeis.org

1, 2, 4, 6, 11, 11, 22, 22, 31, 32, 56, 39, 79, 65, 74, 86, 137, 92, 172, 116, 151, 167, 254, 151, 261, 236, 274, 237, 407, 221, 466, 342, 389, 410, 452, 336, 667, 515, 550, 452, 821, 452, 904, 611, 641, 761, 1082, 599, 1051, 782, 956, 864, 1379, 821, 1166

Offset: 1

Henry Gould, Oct 15 2000

Sum of numerators of n-th order Farey series (cf. A006842). - Benoit Cloitre, Oct 28 2002
Equals row sums of triangle A143613. - Gary W. Adamson, Aug 27 2008
Equals row sums of triangle A159936. - Gary W. Adamson, Apr 26 2009
Also row sums of triangle A164306. - Reinhard Zumkeller, Aug 12 2009

H. W. Gould and Temba Shonhiwa, Functions of GCD's and LCM's, Indian J. Math. (Allahabad), 39 (1997), 11-35.
H. W. Gould and Temba Shonhiwa, A generalization of Cesaro's function and other results, Indian J. Math. (Allahabad), 39 (1997), 183-194.

T. D. Noe, Table of n, a(n) for n = 1..1000
Zachary Franco, Problem 12114, The American Mathematical Monthly, Vol. 126, No. 5 (2019), p. 469; A Dirichlet Series with Reduced Numerators, Solution to Problem 12114 by Tamas Wiandt, ibid., Vol. 128, No. 1 (2021), pp. 91-92.
Index entries for sequences related to lcm's.

Cf. A000010, A000217, A006842, A018804, A023896, A051193, A057660, A143613, A159936, A164306.

See A341316 for another version.

a057661 n = a051193 n `div` n  -- Reinhard Zumkeller, Jun 10 2015

[&+[&+[h: h in [1..d] | GCD(h,d) eq 1]: d in Divisors(n)]: n in [1..100]]; // Jaroslav Krizek, Dec 28 2016

Table[Total[Numerator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; Array[a, 100] (* Amiram Eldar, Apr 26 2023 *)

a(n)=sum(k=1,n,lcm(n,k))/n \\ Charles R Greathouse IV, Feb 07 2017

from math import lcm
def A057661(n): return sum(lcm(n,k)//n for k in range(1,n+1)) # Chai Wah Wu, Aug 24 2023

from math import prod
from sympy import factorint
def A057661(n): return 1+prod((p**((e<<1)+1)+1)//(p+1) for p,e in factorint(n).items())>>1 # Chai Wah Wu, Aug 05 2024

a(n) = (1+A057660(n))/2.
a(n) = A051193(n)/n.
a(n) = Sum_{d|n} psi(d), where psi(m) = is the sum of totatives of m (A023896). - Jaroslav Krizek, Dec 28 2016
a(n) = Sum_{i=1..n} denominator(n/i). - Wesley Ivan Hurt, Feb 26 2017
G.f.: x/(2*(1 - x)) + (1/2)*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, Aug 31 2017
If p is prime, then a(p) = T(p-1) + 1 = p(p-1)/2 + 1, where T(n) = n(n+1)/2 is the n-th triangular number (A000217). - David Terr, Feb 10 2019
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 / Pi^2. - Vaclav Kotesovec, May 29 2021
Dirichlet g.f.: zeta(s)*(1 + zeta(s-2)/zeta(s-1))/2 (Franco, 2019). - Amiram Eldar, Mar 26 2022

More terms from James Sellers, Oct 16 2000

A061255 Euler transform of Euler totient function phi(n), cf. A000010.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 21, 37, 60, 98, 157, 251, 392, 612, 943, 1439, 2187, 3293, 4930, 7330, 10839, 15935, 23315, 33933, 49170, 70914, 101861, 145713, 207638, 294796, 417061, 588019, 826351, 1157651, 1616849, 2251623, 3126775, 4330271, 5981190

Offset: 0

Vladeta Jovovic, Apr 21 2001

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Alessandro De Luca, Gabriele Fici, and Andrea Frosini, Digital Convexity and Combinatorics on Words, arXiv:2502.19926 [math.CO], 2025. See p. 11.
N. J. A. Sloane, Transforms

Cf. A000010, A057660, A006171, A001970, A061256, A061257, A299069, A318975.

nn = 20; b = Table[EulerPhi[n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)

G.f.: Product_{k>=1} (1 - x^k)^(-phi(k)).
a(n) = 1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k) = Sum_{d|k} d*phi(d), cf. A057660.
Logarithmic derivative yields A057660 (equivalent to above formula). - Paul D. Hanna, Sep 05 2012
a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2^(1/3) * Pi^(2/3)) - 1/6) * A^2 * Zeta(3)^(1/9) / (2^(4/9) * 3^(7/18) * Pi^(8/9) * n^(11/18)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 23 2018
G.f.: exp(Sum_{k>=1} (sigma_2(k^2)/sigma_1(k^2)) * x^k/k). - Ilya Gutkovskiy, Apr 22 2019

A054531 Triangular array T read by rows: T(n,k) = n/gcd(n,k) (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 2, 4, 1, 5, 5, 5, 5, 1, 6, 3, 2, 3, 6, 1, 7, 7, 7, 7, 7, 7, 1, 8, 4, 8, 2, 8, 4, 8, 1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 12, 6, 4, 3, 12, 2, 12, 3, 4, 6, 12, 1, 13, 13, 13, 13, 13

Offset: 1

N. J. A. Sloane, Apr 09 2000

Sum of n-th row = A057660(n). - Reinhard Zumkeller, Aug 12 2009
Read as a linear sequence, this is conjectured to be the length of the shortest cycle of pebble-moves among the partitions of n (cf. A201144). - Andrew V. Sutherland, Nov 27 2011
The triangle of fractions A226314(i,j)/A054531(i,j) is an efficient way to enumerate the rationals [Fortnow]. - N. J. A. Sloane, Jun 09 2013

			Triangle begins
   1;
   2,  1;
   3,  3,  1;
   4,  2,  4,  1;
   5,  5,  5,  5,  1;
   6,  3,  2,  3,  6,  1;
   7,  7,  7,  7,  7,  7,  1;
   8,  4,  8,  2,  8,  4,  8,  1;
   9,  9,  3,  9,  9,  3,  9,  9,  1;
  10,  5, 10,  5,  2,  5, 10,  5, 10,  1;
  11, 11, 11, 11, 11, 11, 11, 11, 11, 11,  1;
  12,  6,  4,  3, 12,  2, 12,  3,  4,  6, 12,  1;
  13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,  1;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
R. J. Mathar, Plots of cycle graphs of the finite groups up to order 36, (2015).
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.

Cf. A050873, A164306, A226314, A277227 (row reversed, k=0..n-1).

a054531 n k = div n $ gcd n k
a054531_row n = a054531_tabl !! (n-1)
a054531_tabl = zipWith (\u vs -> map (div u) vs) [1..] a050873_tabl
-- Reinhard Zumkeller, Jun 10 2013

Table[n/GCD[n,k], {n,1,10}, {k,1,n}]//Flatten (* G. C. Greubel, Sep 13 2017 *)

for(n=1,10, for(k=1,n, print1(n/gcd(n,k), ", "))) \\ G. C. Greubel, Sep 13 2017

A066443 Number of distinct paths of length 2n+1 along edges of a unit cube between two fixed adjacent vertices.

Original entry on oeis.org

1, 7, 61, 547, 4921, 44287, 398581, 3587227, 32285041, 290565367, 2615088301, 23535794707, 211822152361, 1906399371247, 17157594341221, 154418349070987, 1389765141638881, 12507886274749927, 112570976472749341

Offset: 0

John W. Layman, Aug 12 2002

All members of sequence are also hex, or central hexagonal, numbers (A003215). (If n is a hex number, 9n - 2 is always a hex number; see recurrence.) - Matthew Vandermast, Mar 30 2003
The sequence 1,1,7,61,547,... with g.f. (1-9x+6x^2)/((1-x)(1-9x)) and a(n) = A054879(n)/3 + 2*0^n/3 gives the denominators in the probability that a random walk on the cube returns to its starting corner on the 2n-th step. - Paul Barry, Mar 11 2004
Equals row sums of even row terms of triangle A158303. - Gary W. Adamson, Mar 15 2009
It appears that a(n) is the n-th record value in A120437, which gives the  differences of A037314 (positive integers n such that the sum of the base 3 digits of n equals the sum of the base 9 digits of n). - John W. Layman, Dec 14 2010
Numbers in base 9 are 1, 6+1, 66+1, 666+1, 6666+1, 66666+1, etc.; that is, n 6's + 1. - Yuchun Ji, Aug 15 2019
All prime factors of a(n) are 1 mod 6. In addition, if n is not 1 mod 3 (first index being n=0), then 3 is a cubic residue modulo all prime factors of a(n). This provides a simple proof that there are infinitely many primes 1 mod 6 that have 3 as a cubic residue. - William Hu, Jul 26 2024

			From _Michael B. Porter_, Aug 22 2016: (Start)
Give coordinates (a,b,c) to the vertices of the cube, where a, b, and c are either 0 or 1. For n = 1, the a(1) = 7 paths of length 2n + 1 = 3 from (0,0,0) to (0,0,1) are:
(0,0,0) -> (0,0,1) -> (0,0,0) -> (0,0,1)
(0,0,0) -> (0,0,1) -> (0,1,1) -> (0,0,1)
(0,0,0) -> (0,0,1) -> (1,0,1) -> (0,0,1)
(0,0,0) -> (0,1,0) -> (0,0,0) -> (0,0,1)
(0,0,0) -> (0,1,0) -> (0,1,1) -> (0,0,1)
(0,0,0) -> (1,0,0) -> (0,0,0) -> (0,0,1)
(0,0,0) -> (1,0,0) -> (1,0,1) -> (0,0,1) (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
G. Benkart and D. Moon, A Schur-Weyl Duality Approach to Walking on Cubes, arXiv preprint arXiv:1409.8154 [math.RT], 2014 and Ann. Combin. 20 (3) (2016) 397-417.
E. Estrada and J. A. de la Pena, From Integer Sequences to Block Designs via Counting Walks in Graphs, arXiv preprint arXiv:1302.1176 [math.CO], 2013. - _N. J. A. Sloane_, Feb 28 2013
E. Estrada and J. A. de la Pena, Integer sequences from walks in graphs, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84.
M. Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly, 54:369-391, 1947.
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 5.
Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A158303, A037314, A120437, A083234 (binomial transform), A083233 (inverse binomial transform), A054879 (recurrent walks), A125857 (walks ending on face diagonal), A054880 (walks ending on space diagonal).

[(3^(2*n+1)+1)/4: n in [0..20]]; // Vincenzo Librandi, Jun 16 2011

seq((3^(2*n+1) + 1)/4, n=0..18); # Zerinvary Lajos, Jun 16 2007

NestList[9 # - 2 &, 1, 18] (* or *)
Table[(3^(2 n + 1) + 1)/4, {n, 0, 18}] (* or *)
CoefficientList[Series[(1 - 3 x)/((1 - x) (1 - 9 x)), {x, 0, 18}], x] (* Michael De Vlieger, Aug 22 2016 *)

a(n)=3^(2*n+1)\/4 \\ Charles R Greathouse IV, Jul 02 2013

Vec((1-3*x)/((1-x)*(1-9*x)) + O(x^50)) \\ Altug Alkan, Nov 13 2015

a(n) = (3^(2*n+1)+1)/4. - Vladeta Jovovic, Dec 22 2002
a(n) = 9*a(n-1) - 2. - Matthew Vandermast, Mar 30 2003
From Paul Barry, Apr 21 2003: (Start)
G.f.: (1-3*x)/((1-x)*(1-9*x)).
E.g.f.: (3*exp(9*x) + exp(x))/4. (End)
a(n) = (-1)^n times the (i, i)-th element of M^n (for any i), where M = ((1, 1, 1, -2), (1, 1, -2, 1), (1, -2, 1, 1), (-2, 1, 1, 1)). - Simone Severini, Nov 25 2004
a(n) = Sum_{k=0..n} binomial(2*n+1, 2*k)*4^(n-k). - Paul Barry, Jan 22 2005
a(n) = A054880(n) + 1.
a(n) = A057660(3^n). - Henry Bottomley, Nov 08 2015
a(n) = Sum_{k=0..2n} (-3)^k == 1 + Sum_{k=1..n} 2*3^(2k-1). - Bob Selcoe, Aug 21 2016
a(n) = 3^(2*n+1) * a(-1-n) for all n in Z. - Michael Somos, Jul 02 2017
a(n) = 6*A002452(n) + 1. - Yuchun Ji, Aug 15 2019

Corrected by Vladeta Jovovic, Dec 22 2002

A068963 a(n) = Sum_{d|n} phi(d^3).

Original entry on oeis.org

1, 5, 19, 37, 101, 95, 295, 293, 505, 505, 1211, 703, 2029, 1475, 1919, 2341, 4625, 2525, 6499, 3737, 5605, 6055, 11639, 5567, 12601, 10145, 13627, 10915, 23549, 9595, 28831, 18725, 23009, 23125, 29795, 18685, 49285, 32495, 38551, 29593

Offset: 1

Benoit Cloitre, Apr 06 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000010, A057660, A056789.

Table[Total[EulerPhi[Divisors[n]^3]],{n,50}] (* Harvey P. Dale, Feb 24 2013 *)
f[p_, e_] := p^2*(p - 1)*(p^(3 e) - 1)/(p^3 - 1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Jun 19 2022 *)

a(n) = sumdiv(n, d, eulerphi(d^3)); \\ Michel Marcus, Mar 10 2018

Also Sum_{d|n} d*phi(d^2), or Sum_{d|n} d^2*phi(d).
Also Sum_{k=1..n} (n/gcd(n, k))^2 = Sum_{k=1..n} (lcm(n, k)/k)^2. - Vladeta Jovovic, Dec 29 2002
Multiplicative with a(p^e) = 1 + p^2 * (p-1)*(p^(3e)-1)/(p^3-1).
G.f.: Sum_{k>=1} k^2*phi(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 10 2018
Dirichlet g.f.: Sum_{n>=1} a(n) / n^s = zeta(s) * zeta(s-3) / zeta(s-2). - Werner Schulte, Feb 18 2021
Sum_{k=1..n} a(k) ~ Pi^2 * n^4 / 60. - Vaclav Kotesovec, Aug 20 2021

cp's OEIS Frontend

A174405 Partial sums of Sum_{k=1..n} n/gcd(n,k), or partial sums of Sum_{d|n} d*phi(d) (see A057660).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A308471 Lowest outliers for A057660.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A318444 Numerators of the sequence whose Dirichlet convolution with itself yields A057660(n) = Sum_{k=1..n} n/gcd(n,k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A377585 E.g.f.: exp(Sum_{k>=1} A057660(k) * x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2*p_i + 3*p_i^2 + ... + (e_i+1)*p_i^e_i).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

PARI

Sage

Formula

Extensions

A057661 a(n) = Sum_{k=1..n} lcm(n,k)/n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Python

Python

Formula

Extensions

A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2p_i + 3p_i^2 + ... + (e_i+1)*p_i^e_i).