A127473 -id:A127473 - cp's OEIS frontend

A358714 a(n) = phi(n)^3.

1, 1, 8, 8, 64, 8, 216, 64, 216, 64, 1000, 64, 1728, 216, 512, 512, 4096, 216, 5832, 512, 1728, 1000, 10648, 512, 8000, 1728, 5832, 1728, 21952, 512, 27000, 4096, 8000, 4096, 13824, 1728, 46656, 5832, 13824, 4096, 64000, 1728, 74088, 8000, 13824, 10648, 97336, 4096, 74088

Offset: 1

Param Mayurkumar Parekh and Paavan Mayurkumar Parekh, Nov 28 2022

Number of solutions to gcd(x*y*z, n) = 1 such that 0 <= x,y,z <= n-1.

x*y*z == t (mod n) where t is a unit (invertible element) in Z_n. Since t is a unit, all x,y,z must be units. Here there are A000010(n) possibilities for each x,y,z so there are a total of A000010(n)^3 ways to get t as a unit.

			a(9) = A000010(9)^3 = 216.

Cf. A000010, A127473, A361148, A335818.

Magma
```
[(EulerPhi(n))^3: n in [1..180]];
```

a[n_] := EulerPhi[n]^3; Array[a, 100] (* Amiram Eldar, Jan 06 2023 *)

PARI
```
a(n) = eulerphi(n)^3;
```

a(n) = A000010(n)^3.
From Amiram Eldar, Jan 06 2023: (Start)
Multiplicative with a(p^e) = (p-1)^3*p^(3*e-3).
Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime}(1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.08429696844... .
Sum_{k>=1} 1/a(k) = Product_{p prime} (1 + p^3/((p-1)^3*(p^3-1))) = 2.47619474816... (A335818). (End)

A377507 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^2 * x^k/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 2, 12, 66, 690, 4860, 63000, 711900, 8876700, 131405400, 2160219600, 37553808600, 686750664600, 13805424032400, 278759396916000, 6445702905642000, 150985820419434000, 3825993309462324000, 99427990563910008000, 2724045313186016820000, 78032929885709378580000

Offset: 0

Vaclav Kotesovec, Oct 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..435

Cf. A318917, A377508, A377509.

Cf. A065464, A127473, A156302.

nmax = 25; $RecursionLimit->Infinity; a[n_]:=a[n]=If[n==0, 1, Sum[EulerPhi[k]^2*a[n-k], {k, 1, n}]/n];Table[a[n]*n!, {n, 0, nmax}]
nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]^2 * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

log(a(n)/n!) ~ 3 * c^(1/3) * n^(2/3) / 2^(2/3), where c = Product_{p primes} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298...

A126775 a(n) = phi(n)^2 * d(n) = A000010(n)^2 * A000005(n).

Original entry on oeis.org

1, 2, 8, 12, 32, 16, 72, 64, 108, 64, 200, 96, 288, 144, 256, 320, 512, 216, 648, 384, 576, 400, 968, 512, 1200, 576, 1296, 864, 1568, 512, 1800, 1536, 1600, 1024, 2304, 1296, 2592, 1296, 2304, 2048, 3200, 1152, 3528, 2400, 3456, 1936, 4232, 2560, 5292, 2400

Offset: 1

Jonathan Vos Post, May 27 2007

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A000010, A064840, A127473, A256392.

[ EulerPhi(n)*EulerPhi(n)*NumberOfDivisors(n) : n in [1..100] ];

Table[EulerPhi[n]^2 DivisorSigma[0,n],{n,50}] (* Harvey P. Dale, Dec 05 2012 *)

Multiplicative with a(p^e) = (e+1)*(p-1)^2*p^(2*e-2). - Amiram Eldar, Dec 29 2022
From Vaclav Kotesovec, May 31 2024: (Start)
Dirichlet g.f.: zeta(s-2)^2 * Product_{p prime} (1 - 1/p^(2*s-2) + 2/p^(2*s-3) - 4/p^(s-1) + 2/p^s).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s-2) + 2/p^(2*s-3) - 4/p^(s-1) + 2/p^s).
Sum_{k=1..n} a(k) ~ f(3) * n^3 * (log(n) + 2*gamma - 1/3 + f'(3)/f(3)) / 3, where
f(3) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264...,
f'(3) = f(3) * Sum_{p prime} 2*(2*p - 1) * log(p) / (p^3 + p^2 - 3*p + 1) = f(3) * 1.6860441157206199528397247528679297282000614932962665074593283751342385...
and gamma is the Euler-Mascheroni constant A001620. (End)

A160620 a(n) = Sum_{d|n} phi(n/d)^2*2^d.

Original entry on oeis.org

0, 2, 6, 16, 28, 64, 96, 200, 320, 616, 1152, 2248, 4304, 8480, 16728, 33152, 66048, 131584, 263160, 524936, 1050176, 2098240, 4196952, 8389576, 16782976, 33555744, 67117920, 134220712, 268453360, 536872480, 1073780352, 2147485448, 4295034880, 8589944384

Offset: 0

N. J. A. Sloane, Nov 21 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A053635.

Cf. A000079, A127473.

A160620 := proc(n)
    if n =0 then
        0;
    else
        add((numtheory[phi](n/d))^2*2^d,d=numtheory[divisors](n)) ;
    end if ;
end proc: # R. J. Mathar, Jun 24 2021

a[n_]:= If[n<1, 0, Sum[EulerPhi[n/d]^2 * 2^d, {d, Divisors[n]}]]; Table[a[n], {n,0,50}] (* G. C. Greubel, May 06 2018 *)

a(n) = if (n, sumdiv(n, d, eulerphi(n/d)^2*2^d), 0); \\ Michel Marcus, May 07 2018, Jun 22 2021

Dirichlet (convolution) product of A127473 and A000079. - R. J. Mathar, Jun 24 2021

A245497 a(n) = phi(n)^2/2, where phi(n) = A000010(n), the Euler totient function.

Original entry on oeis.org

2, 2, 8, 2, 18, 8, 18, 8, 50, 8, 72, 18, 32, 32, 128, 18, 162, 32, 72, 50, 242, 32, 200, 72, 162, 72, 392, 32, 450, 128, 200, 128, 288, 72, 648, 162, 288, 128, 800, 72, 882, 200, 288, 242, 1058, 128, 882, 200, 512, 288, 1352, 162, 800, 288, 648, 392, 1682

Offset: 3

Wesley Ivan Hurt, Jul 24 2014

Values of a(n) < 3 are non-integers since phi(1) = phi(2) = 1 (odd). Since phi(n) is even for all n > 2, a(n) is a positive integer.
a(n) gives the sum of all the parts in the partitions of phi(n) with exactly two parts (see example).
a(n) is also the area of a square with diagonal phi(n).

			a(5) = 8; since phi(5)^2/2 = 4^2/2 = 8. The partitions of phi(5) = 4 into exactly two parts are: (3,1) and (2,2). The sum of all the parts in these partitions gives: 3+1+2+2 = 8.
a(7) = 18; since phi(7)^2/2 = 6^2/2 = 18. The partitions of phi(7) = 6 into exactly two parts are: (5,1), (4,2) and (3,3). The sum of all the parts in these partitions gives: 5+1+4+2+3+3 = 18.

Jens Kruse Andersen, Table of n, a(n) for n = 3..10000

Cf. A000010, A065464, A127473.

with(numtheory): 245497:=n->phi(n)^2/2: seq(245497(n), n=3..50);

Mathematica
```
Table[EulerPhi[n]^2/2, {n, 3, 50}]
```

vector(100, n, eulerphi(n+2)^2/2) \\ Derek Orr, Aug 04 2014

a(n) = phi(n)^2/2 = A000010(n)^2/2 = A127473(n)/2, n > 2.

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/6) * Product_{p prime} (1 - (2*p-1)/p^3) = A065464 / 6 = 0.0713749... . - Amiram Eldar, Nov 14 2024

A279912 a(n) = Sum_{i=1..n} denominator(i^n/n).

Original entry on oeis.org

0, 1, 3, 7, 10, 21, 21, 43, 36, 57, 63, 111, 70, 157, 129, 147, 136, 273, 171, 343, 210, 301, 333, 507, 252, 505, 471, 495, 430, 813, 441, 931, 528, 777, 819, 903, 570, 1333, 1029, 1099, 756, 1641, 903, 1807, 1110, 1197, 1521, 2163, 952, 2065, 1515, 1911, 1570, 2757

Offset: 0

Wesley Ivan Hurt, Dec 22 2016

Multiplicative because this sequence is the Dirichlet convolution of A000027 and A127473 which are both multiplicative. - Andrew Howroyd, Aug 05 2018

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000010, A000027, A065463, A127473, A279911.

List([0..10],n->Sum([1..n],k->DenominatorRat(k^n/n))); # Muniru A Asiru, Oct 24 2018

A279912:=n->add(denom(i^n/n), i=1..n): seq(A279912(n), n=0..100);

Table[DivisorSum[n, # EulerPhi[n/#]^2 &], {n, 53}] (* Michael De Vlieger, Aug 05 2018 *)

a(n) = sum(i=1, n, denominator(i^n/n)); \\ Michel Marcus, Jun 18 2018

a(n) = sumdiv(n, d, d*eulerphi(n/d)^2); \\ Michel Marcus, Jun 18 2018

a(n) = my(f=factor(n)); if(n==0, 0, prod(k=1, #f~, f[k,1]^(f[k,2]-1)  * ((f[k,1]-1) * f[k,1]^f[k,2] + 1))); \\ Daniel Suteu, Oct 24 2018

G.f.: Sum_{k>=1} phi(k)^2*x^k/(1 - x^k)^2, where phi(k) is the Euler totient function. - Ilya Gutkovskiy, Jan 02 2017
a(n) = Sum_{k=1..n} gcd(n, k) * phi(n / gcd(n, k)), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 15 2018
a(n) = Sum_{d|n} d * phi(n/d)^2, where phi(k) is the Euler totient function. - Daniel Suteu, Jun 17 2018
Multiplicative with a(p^k) = p^(k-1) * ((p-1) * p^k + 1). - Daniel Suteu, Oct 24 2018
a(n) = Sum_{k=1..n} n/gcd(n,k)*phi(gcd(n,k))^2/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (1 - 1/(p*(p+1))) = A065463 / 3 = 0.234814... . - Amiram Eldar, Oct 23 2022

A353435 Array read by descending antidiagonals: T(n,m) is the number of sequences of length n >= 0 with elements in 0..m-1 such that the Hankel matrix of any odd number of consecutive terms is invertible over the ring of integers modulo m >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 0, 1, 1, 4, 4, 4, 0, 1, 1, 2, 16, 0, 4, 0, 1, 1, 6, 4, 48, 0, 0, 0, 1, 1, 4, 36, 0, 144, 0, 0, 0, 1, 1, 6, 16, 180, 0, 320, 0, 0, 0, 1, 1, 4, 36, 0, 900, 0, 720, 0, 0, 0, 1, 1, 10, 16, 108, 0, 3744, 0, 1312, 0, 0, 0, 1

Offset: 0

Pontus von Brömssen, Apr 21 2022

T(n,m) is divisible by T(2,m) = A127473(n) for n >= 2, because if r and s are coprime to m, the sequence (x_1, ..., x_n) satisfies the conditions if and only if the sequence (r*s^0*x_1 mod m, ..., r*s^(n-1)*x_n mod m) does.

			Array begins:
  n\m| 1  2  3  4    5  6        7  8   9 10
  ---+--------------------------------------
   0 | 1  1  1  1    1  1        1  1   1  1
   1 | 1  1  2  2    4  2        6  4   6  4
   2 | 1  1  4  4   16  4       36 16  36 16
   3 | 1  0  4  0   48  0      180  0 108  0
   4 | 1  0  4  0  144  0      900  0 324  0
   5 | 1  0  0  0  320  0     3744  0   0  0
   6 | 1  0  0  0  720  0    15552  0   0  0
   7 | 1  0  0  0 1312  0    54216  0   0  0
   8 | 1  0  0  0 2400  0   189468  0   0  0
   9 | 1  0  0  0 3232  0   550728  0   0  0
  10 | 1  0  0  0 4560  0  1604088  0   0  0
  11 | 1  0  0  0 4656  0  3895560  0   0  0
  12 | 1  0  0  0 4928  0  9467856  0   0  0
  13 | 1  0  0  0 4368  0 19185516  0   0  0

Cf. A035287, A045991, A350364, A353433, A353436.
Rows: A000012 (n=0), A000010 (n=1), A127473 (n=2).
Columns: A000012 (m=1), A130716 (m=2), A166926 (m=4 and m=6).

For fixed n, T(n,m) is multiplicative with T(n,p^e) = T(n,p)*p^(n*(e-1)).
T(n,m) = A353436(n,m) if m is prime.
T(3,m) = (m-1)^2*(m-2) = A045991(m-1) if m is prime.
T(4,m) = (m-1)^2*(m-2)^2 = A035287(m-1) if m is prime.
Empirically: T(5,m) = (m-1)^2*(m-3)*(m^2-4*m+5) if m >= 3 is prime.
T(n,2) = 0 for n >= 3.
T(n,3) = 0 for n >= 5.
T(n,5) = 0 for n >= 23.

A049454 a(n) = 1 + Sum_{i=1..n} phi(i)^2.

Original entry on oeis.org

1, 2, 3, 7, 11, 27, 31, 67, 83, 119, 135, 235, 251, 395, 431, 495, 559, 815, 851, 1175, 1239, 1383, 1483, 1967, 2031, 2431, 2575, 2899, 3043, 3827, 3891, 4791, 5047, 5447, 5703, 6279, 6423, 7719, 8043, 8619, 8875, 10475, 10619, 12383

Offset: 0

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000010 (phi), A005728, A057434, A127473.

Table[1+Sum[EulerPhi[i]^2,{i,n}],{n,0,50}] (* Harvey P. Dale, Mar 21 2020 *)
Join[{1}, 1 + Accumulate[EulerPhi[Range[45]]^2]] (* Amiram Eldar, Dec 01 2024 *)

a(n) = 1 + sum(i=1, n, eulerphi(i)^2); \\ Michel Marcus, Mar 07 2020

a(n) = A057434(n) + 1 for n >= 1. - Amiram Eldar, Dec 01 2024

A068484 Numbers k that divide phi(k)^2 + sigma(k)^2.

Original entry on oeis.org

1, 2, 10, 45, 65, 180, 212, 222, 369, 588, 810, 864, 1274, 1521, 1836, 2548, 2940, 3114, 3552, 4770, 5496, 5684, 6027, 6642, 8820, 9140, 10464, 10614, 13311, 14688, 15210, 20737, 21600, 22776, 26900, 27000, 27270, 28476, 28518, 42212, 42336

Offset: 1

Benoit Cloitre, Mar 10 2002

a(275) > 7*10^7. - G. C. Greubel, Oct 15 2018

G. C. Greubel, Table of n, a(n) for n = 1..274

Cf. A072861 (sigma(n)^2), A127473 (phi(n)^2).

Filtered([1..42500],n->(Phi(n)^2+Sigma(n)^2) mod n=0); # Muniru A Asiru, Oct 16 2018

with(numtheory): select(n->modp(phi(n)^2+sigma(n)^2,n)=0,[$1..42500]); # Muniru A Asiru, Oct 16 2018

Select[Range[7000], IntegerQ[(EulerPhi[#]^2 + DivisorSigma[1, #]^2)/#] &] (* G. C. Greubel, Oct 15 2018 *)

A082954 Numbers k such that phi(k*sigma(k)) = phi(k)^2.

Original entry on oeis.org

1, 3, 39, 111, 175, 183, 219, 333, 459, 471, 549, 579, 657, 831, 939, 969, 1191, 1263, 1371, 1413, 1443, 1623, 1737, 1839, 1983, 2019, 2199, 2271, 2379, 2493, 2631, 2817, 2847, 2907, 2991, 3279, 3459, 3573, 3603, 3639, 3711, 3789, 3963, 4113, 4131, 4143

Offset: 1

Benoit Cloitre, May 26 2003

nonn

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

Cf. A000010, A000203, A127473.

Select[Range[5000],EulerPhi[#*DivisorSigma[1,#]]==EulerPhi[#]^2&] (* Harvey P. Dale, Jan 25 2013 *)

cp's OEIS Frontend

A358714 a(n) = phi(n)^3.

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A377507 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^2 * x^k/k), where phi is the Euler totient function A000010.

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A126775 a(n) = phi(n)^2 * d(n) = A000010(n)^2 * A000005(n).

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Magma

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A160620 a(n) = Sum_{d|n} phi(n/d)^2*2^d.

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Maple

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A245497 a(n) = phi(n)^2/2, where phi(n) = A000010(n), the Euler totient function.

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Maple

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A279912 a(n) = Sum_{i=1..n} denominator(i^n/n).

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A353435 Array read by descending antidiagonals: T(n,m) is the number of sequences of length n >= 0 with elements in 0..m-1 such that the Hankel matrix of any odd number of consecutive terms is invertible over the ring of integers modulo m >= 1.

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Comments

Examples

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Formula

A049454 a(n) = 1 + Sum_{i=1..n} phi(i)^2.