cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).

Original entry on oeis.org

1, 5, 13, 25, 41, 65, 85, 121, 157, 205, 221, 325, 313, 425, 533, 569, 545, 785, 685, 1025, 1105, 1105, 1013, 1573, 1441, 1565, 1777, 2125, 1625, 2665, 1861, 2617, 2873, 2725, 3485, 3925, 2665, 3425, 4069, 4961, 3281, 5525, 3613, 5525, 6437, 5065, 4325, 7397, 5965
Offset: 1

Views

Author

Seiichi Manyama, May 21 2024

Keywords

Links

Crossrefs

Row sums of A350900.
Cf. A057660, A350156.
Cf. A001620, A002117, A013661, A073002, A244115, A306016.

Programs

  • Mathematica
    f[p_, e_] := (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 27 2024 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, gcd(i, n)/gcd([i, j, n])));
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2);} \\ Amiram Eldar, May 27 2024

Formula

a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_2(d^2)/sigma(d^2).
From Amiram Eldar, May 27 2024: (Start)
Multiplicative with a(p^e) = (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2.
Dirichlet g.f.: zeta(s) * zeta(s-2)^2 / zeta(s-1)^2.
Sum_{k=1..n} a(k) ~ (2*zeta(3)*n^3/(15*zeta(4))) * (log(n) + 2*gamma - 1/3 - 2*zeta'(2)/zeta(2) + zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)

A061259 a(n)=Sum_{d|n} d*numbpart(d), where numbpart(d)=number of partitions of d, cf. A000041.

Original entry on oeis.org

1, 5, 10, 25, 36, 80, 106, 201, 280, 460, 617, 1024, 1314, 2000, 2685, 3897, 5050, 7280, 9311, 13020, 16747, 22665, 28866, 39000, 48986, 64654, 81550, 106124, 132386, 171295, 212103, 271065, 335345, 423594, 521046, 655396, 800570, 997885
Offset: 1

Views

Author

Vladeta Jovovic, Apr 21 2001

Keywords

Links

Crossrefs

Cf. A000041, A057660, A001001, A060640, A061258.
Cf. A001970, A027750.

Programs

  • Haskell
    a061259 n = sum $ zipWith (*) divs $ map a000041 divs
            where divs = a027750_row' n
-- Reinhard Zumkeller, Oct 31 2015

A063445 Moebius transform of f(x) = EulerPhi(x^2) function (A002618).

Original entry on oeis.org

1, 1, 5, 6, 19, 5, 41, 24, 48, 19, 109, 30, 155, 41, 95, 96, 271, 48, 341, 114, 205, 109, 505, 120, 480, 155, 432, 246, 811, 95, 929, 384, 545, 271, 779, 288, 1331, 341, 775, 456, 1639, 205, 1805, 654, 912, 505, 2161, 480, 2016, 480, 1355, 930, 2755, 432
Offset: 1

Views

Author

Labos Elemer, Jul 24 2001

Keywords

Comments

Same as Moebius transform of g(x) = x*EulerPhi(x). - Benoit Cloitre, Apr 05 2002

Examples

			For n=20, divisors = {1,2,4,5,10,20}, phi(d^2) = {1,2,8,20,40,160}, mu(20/d) = {0,1,-1,0,-1,1}, a(20) = 0 + 2 - 8 + 0 - 40 + 160 = 114.
a(20) = a(4)*a(5) = (16 - 8 - 4 + 2)*(25 - 5 - 1) = 6*19 = 114.

Links

Crossrefs

Cf. A000010, A002618, A057660, A007434.

Programs

  • Mathematica
    Table[Sum[EulerPhi[d]*MoebiusMu[n/d]*d, {d, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, Feb 01 2019 *)
  • PARI
    a(n)=if(n<1,0,sumdiv(n,d,d*eulerphi(d)*moebius(n/d)))

Formula

a(n) = Sum_{d|n} phi(d^2)*mu(n/d).
Multiplicative with a(p) = p^2 - p - 1 and a(p^e) = p^(2*e) - p^(2*e-1) - p^(2*e-2) + p^(2*e-3), e > 1. - Vladeta Jovovic, Jul 29 2001
Dirichlet g.f. zeta(s-2)/(zeta(s)*zeta(s-1)). - R. J. Mathar, Feb 09 2011
Sum_{k=1..n} a(k) ~ 2*n^3 / (Pi^2 * Zeta(3)). - Vaclav Kotesovec, Feb 01 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2-p-1) + p/((p-1)^3 * (p+1)^2)) = 3.037448431566721466562170968413075105160439538735056586164601312913619316... - Vaclav Kotesovec, Sep 20 2020
a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)*moebius(gcd(i, j, n)) = Sum_{d divides n} d*moebius(d)*J_2(n/d), where J_2 is the Jordan totient function A007434. - Peter Bala, Jan 21 2024

A064951 a(n) = Sum_{1 <= x, y <= n} lcm(x, y).

Original entry on oeis.org

1, 7, 28, 72, 177, 303, 604, 948, 1497, 2127, 3348, 4272, 6313, 8119, 10324, 13060, 17701, 20995, 27512, 32132, 38453, 45779, 57440, 64664, 77689, 89935, 104704, 117948, 141525, 154755, 183616, 205472, 231113, 258959, 290564, 314720, 364041
Offset: 1

Views

Author

Vladeta Jovovic, Oct 28 2001

Keywords

Comments

a(n) is also the entrywise 1-norm of the n X n LCM matrix.

Links

Crossrefs

Cf. A018806, A051193, A057660.

Programs

  • Mathematica
    Table[nn = n;Total[Level[Table[Table[LCM[i, j], {i, 1, nn}], {j, 1, nn}], {2}]], {n, 1, 37}] (* Geoffrey Critzer, Jan 14 2015 *)
  • PARI
    { a=0; for (n=1, 1000, a+=n*sum(k=1, n, n/gcd(n, k)); write("b064951.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 01 2009

Formula

a(n) = a(n-1) + 2*A051193(n) - n = a(n-1) + n*A057660(n) = Sum_{1 <= i <= j <= n} (j^2/gcd(i, j)). - Henry Bottomley, Oct 29 2001
a(n) ~ 3 * zeta(3) * n^4 / (2*Pi^2). - Vaclav Kotesovec, May 29 2021

A078747 Expansion of Sum_{k>0} k*phi(k)*x^k/(1+x^k).

Original entry on oeis.org

1, 1, 7, 5, 21, 7, 43, 21, 61, 21, 111, 35, 157, 43, 147, 85, 273, 61, 343, 105, 301, 111, 507, 147, 521, 157, 547, 215, 813, 147, 931, 341, 777, 273, 903, 305, 1333, 343, 1099, 441, 1641, 301, 1807, 555, 1281, 507, 2163, 595, 2101, 521, 1911, 785, 2757, 547
Offset: 1

Views

Author

Vladeta Jovovic, Dec 22 2002

Keywords

Links

Crossrefs

Cf. A000010, A057660, A240976, A299069.

Programs

  • Mathematica
    f[p_, e_] := If[p == 2, (4^e - 1)/3, (p^(2*e + 1) + 1)/(p + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 15 2022 *)
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, (4^f[i,2]-1)/3, (f[i,1]^(2*f[i,2]+1)+1)/(f[i,1]+1))); } \\ Amiram Eldar, Oct 15 2022

Formula

Multiplicative with a(2^e) = (4^e-1)/3, a(p^e) = (p^(2*e+1)+1)/(p+1), p>2.
L.g.f.: log(Product_{k>=1} (1 + x^k)^phi(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/(4*zeta(2)) = 0.182690... (A240976). - Amiram Eldar, Oct 15 2022
Dirichlet g.f.: (zeta(s)*zeta(s-2)/zeta(s-1))*(1-2^(1-s)). - Amiram Eldar, Dec 30 2022

A286946 a(1) = 1; a(n+1) = Sum_{k=1..n} a(n)/gcd(a(k),a(n)).

Original entry on oeis.org

1, 1, 2, 5, 16, 57, 286, 1431, 9064, 51398, 359787, 3118155, 25568872, 223727631, 2311852188, 15990310968, 105935810164, 1038449718056, 10903722039589, 185715007642033, 3528585145198628, 46753753173881822, 658243630211230916, 9215410822957232825, 197209791611284782456, 2112570763708981231112
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 31 2017

Keywords

Examples

			a(1) = 1;
a(2) = a(1)/gcd(a(1),a(1)) = 1/gcd(1,1) = 1;
a(3) = a(2)/gcd(a(1),a(2)) + a(2)/gcd(a(2),a(2)) = 1/gcd(1,1) + 1/gcd(1,1) = 2;
a(4) = a(3)/gcd(a(1),a(3)) + a(3)/gcd(a(2),a(3)) + a(3)/gcd(a(3),a(3)) = 2/gcd(1,2) + 2/gcd(1,2) + 2/gcd(2,2) = 5, etc.

Links

Crossrefs

Cf. A056144, A057660, A093820, A287006.

Programs

  • Maple
    A[1]:= 1:
for n from 1 to 50 do
  A[n+1]:= add(A[n]/igcd(A[k],A[n]),k=1..n)
od:
seq(A[i],i=1..50); # Robert Israel, Sep 01 2017
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[a[n - 1]/GCD[a[k - 1], a[n - 1]], {k, 2, n}]; Table[a[n], {n, 26}]
a[1] = 1; a[n_] := a[n] = Sum[LCM[a[k - 1], a[n - 1]]/a[k - 1], {k, 2, n}]; Table[a[n], {n, 26}]

Formula

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

A332049 a(n) = (1/2) * Sum_{d|n, d > 1} d * phi(d).

Original entry on oeis.org

0, 1, 3, 5, 10, 10, 21, 21, 30, 31, 55, 38, 78, 64, 73, 85, 136, 91, 171, 115, 150, 166, 253, 150, 260, 235, 273, 236, 406, 220, 465, 341, 388, 409, 451, 335, 666, 514, 549, 451, 820, 451, 903, 610, 640, 760, 1081, 598, 1050, 781, 955, 863, 1378, 820, 1165
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 06 2020

Keywords

Comments

Sum of numerators of the reduced fractions 1/n, ..., (n-1)/n. Note that if n is a prime p this is p*(p-1)/2 as all fractions are already reduced. For 1/n, ..., n/n, see A057661.

Examples

			For n = 5, fractions are 1/5, 2/5, 3/5, 4/5, sum of numerators is 10.
For n = 8, fractions are 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, sum of numerators is 21.

Links

Crossrefs

Cf. A000010, A002618, A006579, A023896, A057660, A057661.

Programs

  • Haskell
    toNums a = fmap (numerator . (% a))
toNumList a = toNums a [1..(a-1)]
sumList = sum . toNumList <$> [2..200]
  • Magma
    [0] cat [(1/2)*&+[ d*EulerPhi(d):d in Set(Divisors(n)) diff {1}]:n in [2..60]]; // Marius A. Burtea, Feb 07 2020
  • Maple
    N:= 100: # for a(1)..a(N)
V:= Vector(N):
for d from 2 to N do
  v:= d*numtheory:-phi(d)/2;
  R:= [seq(i,i=d..N,d)];
  V[R]:= V[R] +~ v
od:
convert(V,list); # Robert Israel, Feb 07 2020
  • Mathematica
    Table[(1/2) Sum[If[d > 1, d EulerPhi[d], 0], {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[(1/2) Sum[EulerPhi[k^2] x^k/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[k/GCD[n, k], {k, 1, n - 1}], {n, 1, 55}]
Table[(DivisorSigma[2, n^2] - DivisorSigma[1, n^2])/(2 DivisorSigma[1, n^2]), {n, 1, 55}]
  • PARI
    a(n) = sumdiv(n, d, if (d>1, d*eulerphi(d)))/2; \\ Michel Marcus, Feb 07 2020

Formula

G.f.: (1/2) * Sum_{k>=2} phi(k^2) * x^k / (1 - x^k).
a(n) = Sum_{k=1..n-1} k / gcd(n,k).
a(n) = (sigma_2(n^2) - sigma_1(n^2)) / (2 * sigma_1(n^2)).
a(n) = Sum_{d|n, d > 1} A023896(d).
a(n) = A057661(n) - 1 = (A057660(n) - 1) / 2.

A332792 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(d) * a(d).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 6, 3, 6, 1, 16, 1, 8, 7, 30, 1, 30, 1, 34, 9, 12, 1, 104, 5, 14, 21, 60, 1, 96, 1, 270, 13, 18, 11, 278, 1, 20, 15, 330, 1, 174, 1, 136, 81, 24, 1, 1176, 7, 130, 19, 186, 1, 588, 15, 804, 21, 30, 1, 1204, 1, 32, 135, 4590, 17, 402, 1, 310, 25, 348
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 24 2020

Keywords

Links

Crossrefs

Cf. A000010, A006874, A008578 (positions of 1's), A038045, A057660, A332791.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Sum[If[d < n, EulerPhi[d] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
a[1] = 1; a[n_] := a[n] = Sum[If[GCD[n, k] > 1, a[n/GCD[n, k]], 0], {k, 1, n}]; Table[a[n], {n, 1, 70}]
  • PARI
    up_to = 20000;
A332792list(n) = { my(v=vector(n)); v[1] = 1; for(n=2, #v, v[n] = sumdiv(n, d, if(d==n, 0, v[d]*eulerphi(d)))); (v); };
v332792 = A332792list(up_to);
A332792(n) = v332792[n]; \\ Antti Karttunen, Jan 22 2025

Formula

a(1) = 1; a(n) = Sum_{k=1..n, gcd(n, k) > 1} a(n/gcd(n, k)).

A341316 Row sums in A341315.

Original entry on oeis.org

0, 3, 6, 12, 18, 33, 33, 66, 66, 93, 96, 168, 117, 237, 195, 222, 258, 411, 276, 516, 348, 453, 501, 762, 453, 783, 708, 822, 711, 1221, 663, 1398, 1026, 1167, 1230, 1356, 1008, 2001, 1545, 1650, 1356, 2463, 1356, 2712, 1833, 1923, 2283, 3246, 1797, 3153, 2346, 2868, 2592, 4137
Offset: 0

Views

Author

N. J. A. Sloane, Feb 17 2021

Keywords

Comments

This is three times A057661. See that entry for much more information.

Crossrefs

Cf. A057660, A057661, A341314, A341315.

Formula

a(n) = 3*(A057660(n)+1)/2 for n>=1. - Hugo Pfoertner, Feb 17 2021

A344508 a(n) = Sum_{k=1..n} k * lcm(k,n).

Original entry on oeis.org

1, 6, 24, 64, 175, 270, 686, 928, 1647, 2150, 4356, 3792, 8619, 8526, 11250, 14592, 25721, 19926, 40432, 31200, 44835, 53966, 87814, 58272, 108125, 106470, 132678, 124656, 224547, 132750, 294066, 232960, 284229, 316166, 372400, 291168, 600991, 496014, 560742, 484000, 909421, 531846
Offset: 1

Views

Author

Seiichi Manyama, May 21 2021

Keywords

Links

Crossrefs

Cf. A051193, A057660, A344509, A344510.

Programs

  • Mathematica
    a[n_] := Sum[k * LCM[k, n], {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 21 2021 *)
  • PARI
    a(n) = sum(k=1, n, k*lcm(k, n));

Formula

Sum_{k=1..n} a(k) ~ 2 * zeta(3) * n^5 / (5*Pi^2). - Vaclav Kotesovec, May 29 2021
Previous Showing 31-40 of 65 results. Next

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