A057660 -id:A057660 - cp's OEIS frontend

A061258 a(n) = Sum_{d|n} d*psi(d), where psi(d) is reduced totient function, cf. A002322.

1, 3, 7, 11, 21, 21, 43, 27, 61, 63, 111, 53, 157, 129, 87, 91, 273, 183, 343, 151, 175, 333, 507, 117, 521, 471, 547, 305, 813, 261, 931, 347, 447, 819, 483, 431, 1333, 1029, 631, 327, 1641, 525, 1807, 781, 681, 1521, 2163, 373, 2101, 1563, 1095, 1103, 2757

Offset: 1

Vladeta Jovovic, Apr 21 2001

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002322, A057660, A001001, A060640, A061259.

Cf. A000005, A027750, A141258.

a061258 n = sum $ zipWith (*) ds $ map a002322 ds
            where ds = a027750_row n
-- Reinhard Zumkeller, Sep 02 2014

a[n_] := DivisorSum[n, # * CarmichaelLambda[#] &]; Array[a, 100] (* Amiram Eldar, Apr 13 2024 *)

a(n) = sumdiv(n, d, d * lcm(znstar(d)[2])); \\ Amiram Eldar, Apr 13 2024

a(n) = Sum_{k = 1..A000005(n)} (A027750(n,k)*A002322(A027750(n,k))). - Reinhard Zumkeller, Sep 02 2014

A332619 a(n) = Sum_{d|n} lcm(d, n/d) / d.

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 11, 18, 12, 24, 14, 24, 24, 23, 18, 33, 20, 36, 32, 36, 24, 48, 27, 42, 32, 48, 30, 72, 32, 45, 48, 54, 48, 66, 38, 60, 56, 72, 42, 96, 44, 72, 66, 72, 48, 92, 51, 81, 72, 84, 54, 96, 72, 96, 80, 90, 60, 144, 62, 96, 88, 88, 84, 144, 68, 108, 96, 144

Offset: 1

Ilya Gutkovskiy, Feb 17 2020

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to gcd's.
Index entries for sequences related to lcm's.

Cf. A055155, A057660, A057661, A057670, A332618.

Cf. A013663, A013664.

a:= n-> add(d/igcd(d, n/d), d=numtheory[divisors](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 17 2020

Table[Sum[LCM[d, n/d]/d, {d, Divisors[n]}], {n, 1, 70}]
f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1) + e/2, (p^(e + 2) - p)/(p^2 - 1) + (e + 1)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

A332619(n) = sumdiv(n,d,lcm(d,n/d)/d); \\ Antti Karttunen, Nov 12 2021

a(n) = Sum_{d|n} d / gcd(d, n/d).
From Amiram Eldar, Dec 05 2022: (Start)
Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) + e/2 if e is even, and (p^(e+2)-p)/(p^2-1) + (e + 1)/2 if e is odd.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 7*zeta(6)/(8*zeta(5)) = 0.740543... . (End)

A372968 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} n/gcd(x_1, x_2, ..., x_k, n).

Original entry on oeis.org

1, 1, 3, 1, 7, 7, 1, 15, 25, 11, 1, 31, 79, 55, 21, 1, 63, 241, 239, 121, 21, 1, 127, 727, 991, 621, 175, 43, 1, 255, 2185, 4031, 3121, 1185, 337, 43, 1, 511, 6559, 16255, 15621, 7471, 2395, 439, 61, 1, 1023, 19681, 65279, 78121, 45801, 16801, 3823, 673, 63

Offset: 1

Seiichi Manyama, May 18 2024

			Square array begins:
   1,   1,    1,     1,      1,      1, ...
   3,   7,   15,    31,     63,    127, ...
   7,  25,   79,   241,    727,   2185, ...
  11,  55,  239,   991,   4031,  16255, ...
  21, 121,  621,  3121,  15621,  78121, ...
  21, 175, 1185,  7471,  45801, 277495, ...

Columns k=1..5 give A057660, A350156, A372952, A372961, A371878.

Main diagonal gives A372969.

f[p_, e_, k_] := (p^((k + 1)*e + k + 1) - p^((k + 1)*e + 1) + p - 1)/(p^(k + 1) - 1); T[1, k_] := 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 25 2024 *)

T(n, k) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, k+1));

T(n,k) = Sum_{d|n} mu(n/d) * (n/d) * sigma_{k+1}(d).
T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} ( gcd(x_1, x_2, ..., x_{k-1}, n)/gcd(x_1, x_2, ..., x_k, n) )^k.
From Amiram Eldar, May 25 2024: (Start)
T(n,k) for a given k is multiplicative with T(p^e, k) = (p^((k+1)*(e+1)) - p^((k+1)*e+1) + p - 1)/(p^(k+1)-1).
Dirichlet g.f. of T(n, k) for a given k: zeta(s)*zeta(s-k-1)/zeta(s-1).
Sum_{m=1..n} T(m, k) ~ c * n^(k+2) / (k+2), where c = zeta(k+2)/zeta(k+1). (End)

A373105 a(n) = sigma_10(n^2)/sigma_5(n^2).

Original entry on oeis.org

1, 993, 58807, 1016801, 9762501, 58395351, 282458443, 1041204193, 3472494301, 9694163493, 25937263551, 59795016407, 137858120557, 280481233899, 574103396307, 1066193093601, 2015992480593, 3448186840893, 6131063781703, 9926520779301

Offset: 1

Seiichi Manyama, May 25 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A057660, A084218, A084220, A372966.
Cf. A371491, A371878, A373007, A373103.
Cf. A013664, A013669.

f[p_, e_] := (p^(10*e+5) + 1)/(p^5 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, May 25 2024 *)

PARI
```
a(n) = sigma(n^2, 10)/sigma(n^2, 5);
```

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^5*sigma(d, 10));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x^5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^5 * sigma_10(d).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(10*e+5) + 1)/(p^5 + 1).
Dirichlet g.f.: zeta(s)*zeta(s-10)/zeta(s-5).
Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = zeta(11)/zeta(6) = 0.9834383562... . (End)

A056916 Product of the orders of the elements in a cyclic group with n elements.

Original entry on oeis.org

1, 2, 9, 32, 625, 648, 117649, 131072, 4782969, 12500000, 25937424601, 214990848, 23298085122481, 1771684761728, 14416259765625, 562949953421312, 48661191875666868481, 11712917736940032, 104127350297911241532841, 5120000000000000000, 7788651757984142343081

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Feb 10 2002

Amiram Eldar, Table of n, a(n) for n = 1..388

Cf. A057660.

Cf. A067911, A071248.

a:=n->mul(denom (k/n), k=1..n): seq(a(n), n=1..18); # Zerinvary Lajos, Apr 26 2008

Table[Product[n/GCD[n,i],{i,0,n-1}],{n,30}] (* Harvey P. Dale, Oct 24 2011 *)

a(n) = Product_{ d divides n } d^phi(d). - Vladeta Jovovic, Sep 10 2004

Edited by Dean Hickerson, Mar 04 2002

A077454 a(n) = sigma_3(n^3)/sigma(n^3).

Original entry on oeis.org

1, 39, 511, 2359, 12621, 19929, 101179, 149943, 368089, 492219, 1611831, 1205449, 4457701, 3945981, 6449331, 9588151, 22722609, 14355471, 44576623, 29772939, 51702469, 62861409, 141611691, 76620873, 196890121, 173850339, 268218727, 238681261, 574336533, 251523909

Offset: 1

Benoit Cloitre, Nov 30 2002

			a(2) = sigma_3(2^3)/sigma(2^3) = 585/15 = 39.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A000578, A001158, A013665, A057660, A077455, A077456.

f[p_, e_] := (p^(6*e+2) + p^(3*e+1) + 1)/(p^2 + p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 30] (* Amiram Eldar, Sep 09 2020 *)

PARI
```
a(n)=sumdiv(n^3,d,d^3)/sigma(n^3)
```

a(n) = my(f=factor(n^3)); sigma(f, 3)/sigma(f); \\ Michel Marcus, Sep 09 2020

a(n) = A001158(n^3)/A000203(n^3).
Multiplicative with a(p^e) = (p^(6*e+2) + p^(3*e+1) + 1)/(p^2 + p + 1). - Amiram Eldar, Sep 09 2020
Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)*Pi^4/630) * Product_{p prime} (1 - 1/p^2 - 1/p^6 + 1/p^7 - 1/p^8 + 1/p^9) = 0.09343400455... . - Amiram Eldar, Oct 28 2022

A077455 a(n) = sigma_4(n^4)/sigma(n^4).

Original entry on oeis.org

1, 2255, 360205, 8965359, 195688121, 812262275, 11869610005, 36654862063, 190649623129, 441276712855, 2853329308061, 3229367138595, 21506735660905, 26765970561275, 70487839624805, 150121132912367, 548357292625505, 429914900155895, 2096841596815405, 1754414256800439

Offset: 1

Benoit Cloitre, Nov 30 2002

			a(2) = sigma_4(2^4)/sigma(2^4) = 69905/31 = 2255.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A000583, A001158, A057660, A077454, A077456.

Cf. A002117, A013663, A013667, A013671.

f[p_, e_] := (p^(12*e+3) + p^(8*e+2) + p^(4*e+1) + 1)/(p^3 + p^2 + p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 20] (* Amiram Eldar, Sep 09 2020 *)

PARI
```
a(n)=sumdiv(n^4,d,d^4)/sigma(n^4)
```

a(n) = my(f=factor(n^4)); sigma(f, 4)/sigma(f); \\ Michel Marcus, Sep 09 2020

a(n) = A001158(n^4)/A000203(n^4).
Multiplicative with a(p^e) = (p^(12*e+3) + p^(8*e+2) + p^(4*e+1) + 1)/(p^3 + p^2 + p + 1). - Amiram Eldar, Sep 09 2020
Sum_{k=1..n} a(k) ~ c * n^13, where c = (zeta(3)*zeta(5)*zeta(9)*zeta(13)/13) * Product_{p prime} (1-1/p^2-1/p^3+1/p^5-1/p^7+1/p^8-1/p^12+2/p^13-2/p^14+2/p^15-1/p^16+2/p^17-3/p^18+1/p^19+1/p^21-1/p^22-1/p^26-1/p^27) = 0.048281563902... . - Amiram Eldar, Nov 20 2022

A077456 a(n) = sigma_5(n^5)/sigma(n^5).

Original entry on oeis.org

1, 549791, 2337334621, 567767102431, 76323251878121, 1285045538614211, 68398022066406901, 595065340418751455, 8138648440293876241, 41961836973324022711, 611595047235520833101, 1327061705176829563651, 17543094367661056941241, 37604616949911916507691

Offset: 1

Benoit Cloitre, Nov 30 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A000584, A001160, A057660, A077454, A077455.

f[p_, e_] := (p^(20*e+4) + p^(15*e+3) + p^(10*e+2) + p^(5*e+1) + 1)/(p^4 + p^3 + p^2 + p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 20] (* Amiram Eldar, Sep 09 2020 *)
Table[DivisorSigma[5,n^5]/DivisorSigma[1,n^5],{n,20}] (* Harvey P. Dale, Mar 05 2022 *)

PARI
```
a(n)=sumdiv(n^5,d,d^5)/sigma(n^5)
```

a(n) = my(f=factor(n^5)); sigma(f, 5)/sigma(f); \\ Michel Marcus, Sep 09 2020

a(n) = A001160(n^5)/A000203(n^5).

Multiplicative with a(p^e) = (p^(20*e+4) + p^(15*e+3) + p^(10*e+2) + p^(5*e+1) + 1)/(p^4 + p^3 + p^2 + p + 1). - Amiram Eldar, Sep 09 2020

A332791 a(1) = 1; a(n+1) = Sum_{d|n} phi(d) * a(d).

Original entry on oeis.org

1, 1, 2, 5, 12, 49, 104, 625, 2512, 15077, 60358, 603581, 2414438, 28973257, 173840168, 1390721397, 11125773688, 178012379009, 1068074289230, 19225337206141, 153802697709496, 1845632372514581, 18456323725749392, 406039121966486625, 3248312975734309938

Offset: 1

Ilya Gutkovskiy, Feb 24 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..495

Cf. A000010, A006874, A038045, A057660, A307793, A307794, A332792.

a[1] = 1; a[n_] := Sum[EulerPhi[d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 25}]
a[1] = 1; a[n_] := a[n] = Sum[a[(n - 1)/GCD[n - 1, k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 25}]

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

a(n) = Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A369779 a(n) = n * Sum_{p|n, p prime} phi(n/p) / p.

Original entry on oeis.org

0, 1, 1, 2, 1, 8, 1, 8, 6, 22, 1, 20, 1, 44, 26, 32, 1, 66, 1, 48, 48, 112, 1, 80, 20, 158, 54, 92, 1, 172, 1, 128, 116, 274, 62, 156, 1, 344, 162, 192, 1, 348, 1, 228, 174, 508, 1, 320, 42, 540, 278, 320, 1, 594, 130, 368, 348, 814, 1, 448, 1, 932, 306, 512, 176

Offset: 1

Wesley Ivan Hurt, Jan 31 2024

Dirichlet convolution of A010051(n) and A002618(n). - Wesley Ivan Hurt, Jul 10 2025

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A002618, A010051, A057660, A117494, A347104, A369687.

Cf. also A369894, A369907, A369909.

Table[n*DivisorSum[n, EulerPhi[n/#]/# &, PrimeQ[#] &], {n, 100}]

A369779(n) = if(1==n, 0, my(f=factor(n)); n*sum(i=1, #f~, (eulerphi(n/f[i, 1])/f[i,1]))); \\ Antti Karttunen, Jan 23 2025

From Wesley Ivan Hurt, Jul 10 2025: (Start)
a(n) = Sum_{d|n} A010051(d) * A002618(n/d).
a(p^k) = ceiling(p^(2k-2)-p^(2k-3)) for p prime and k>=1. (End)

cp's OEIS Frontend

A061258 a(n) = Sum_{d|n} d*psi(d), where psi(d) is reduced totient function, cf. A002322.

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A332619 a(n) = Sum_{d|n} lcm(d, n/d) / d.

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A372968 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} n/gcd(x_1, x_2, ..., x_k, n).

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A373105 a(n) = sigma_10(n^2)/sigma_5(n^2).

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A056916 Product of the orders of the elements in a cyclic group with n elements.

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A077454 a(n) = sigma_3(n^3)/sigma(n^3).

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A077455 a(n) = sigma_4(n^4)/sigma(n^4).

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A077456 a(n) = sigma_5(n^5)/sigma(n^5).