A053287 -id:A053287 - cp's OEIS frontend

A295503 a(n) = phi(10^n-1), where phi is Euler's totient function (A000010).

6, 60, 648, 6000, 64800, 466560, 6637344, 58752000, 648646704, 5890320000, 66663457344, 461894400000, 6458084523072, 60339430569600, 610154104320000, 5529599115264000, 66666634474902192, 441994921381739520, 6666666666666666660, 58301444908800000000

Offset: 1

Seiichi Manyama, Nov 22 2017

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352
Eric W. Weisstein, MathWorld: Totient Function
Wikipedia, Euler's totient function

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), this sequence (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A002283.

Array[ EulerPhi[10^# - 1] &, 20] (* Robert G. Wilson v, Nov 22 2017 *)

PARI
```
{a(n) = eulerphi(10^n-1)}
```

a(n) = n*A295497(n).

a(n) = A000010(A002283(n)). - Michel Marcus, Nov 25 2017

A096853 a(n) = A062401(2^n-1).

Original entry on oeis.org

1, 2, 4, 8, 16, 48, 64, 144, 288, 512, 576, 2304, 4096, 10240, 18432, 36288, 65536, 184320, 262144, 552960, 718848, 1492992, 2822400, 9123840, 13418496, 44695552, 68762880, 106168320, 109486080, 580386816, 1073741824, 2155507200, 2366668800, 6920601600, 12081954816

Offset: 1

Labos Elemer, Jul 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..392

Cf. A000010, A000203, A000225, A062401, A062402, A053287, A075708, A096853, A096854, A096855, A096856.

Table[ EulerPhi[ DivisorSigma[1, 2^n - 1]], {n, 33}]

a(n) = A000010(A000203(A000225(n))). - Michel Marcus, Dec 19 2013

a(n) = A000010(A075708(n)). - Amiram Eldar, Jun 04 2024

Edited and extended by Robert G. Wilson v, Jul 23 2004

a(33)-a(35) from Amiram Eldar, Jun 04 2024

A096854 a(n) = A062402(2^n-1).

Original entry on oeis.org

1, 3, 12, 15, 72, 91, 312, 255, 1240, 1860, 4123, 5080, 26208, 34200, 93600, 65535, 334368, 416560, 1420800, 1596364, 6146800, 5949696, 20485332, 23788842, 120519630, 194016600, 358132380, 458803800, 1674738000, 2166798816, 6045990912, 4294967295, 22739738112, 37862623140

Offset: 1

Labos Elemer, Jul 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..400

Cf. A000203, A053287, A062401, A062402, A053287, A096853, A096854, A096855, A096856.

Table[DivisorSigma[1, EulerPhi[2^n - 1]], {n, 1, 30}]

a(n) = sigma(eulerphi(2^n-1)); \\ Michel Marcus, Aug 30 2019

a(n) = A000203(A053287(n)). - Amiram Eldar, Jun 04 2024

More terms from Michel Marcus, Aug 30 2019

A096855 a(n) = A062401(2^n + 1).

Original entry on oeis.org

2, 2, 2, 12, 6, 16, 24, 80, 84, 320, 360, 864, 1320, 5456, 5184, 15744, 19800, 52800, 69120, 349520, 370080, 1036800, 1425600, 3640896, 4741632, 13989888, 27091584, 76743040, 94656000, 166387200, 412473600, 1407389952, 1420488192, 3459760128, 6502788864, 14778408960

Offset: 0

Labos Elemer, Jul 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..400

Cf. A000010, A062401, A062402, A053287, A096853, A069061, A096854, A096855, A096856.

Table[ EulerPhi[ DivisorSigma[1, 2^n+1]], {n, 0, 33}]

a(n) = A000010(A069061(n)). - Amiram Eldar, Jun 04 2024

Edited and extended by Robert G. Wilson v, Jul 23 2004

Offset changed to 0, a(0) prepended and two more terms added by Amiram Eldar, Jun 04 2024

A096856 a(n) = A062402(2^n+1).

Original entry on oeis.org

1, 3, 7, 12, 31, 42, 124, 224, 511, 847, 1953, 2688, 12264, 18816, 29127, 72540, 131071, 195048, 558523, 1077440, 3164112, 4552020, 10890040, 10342080, 54525848, 73260781, 155671040, 318848400, 1080311232, 964580240, 3070642080, 4340711424, 13722819600, 19039027200

Offset: 0

Labos Elemer, Jul 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..372

Cf. A000203, A053285, A062401, A062402, A053287, A096853, A096854, A096855.

Table[DivisorSigma[1, EulerPhi[2^n + 1]], {n, 0, 30}]

a(n) = A000203(A053285(n)). - Amiram Eldar, Jun 04 2024

Offset changed to 0, a(0) prepended and three more terms added by Amiram Eldar, Jun 04 2024

A057764 Triangle T(n,k) = number of nonzero elements of multiplicative order k in Galois field GF(2^n) (n >= 1, 1 <= k <= 2^n-1).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 0, 0, 0, 0, 6, 1, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 1, 0, 2, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36

Offset: 1

N. J. A. Sloane, Nov 01 2000

			Table begins:
  1;
  1, 0, 2;
  1, 0, 0, 0, 0, 0, 6;
  ...

Robert Israel, Table of n, a(n) for n = 1..16369 (rows 1 to 13, flattened)

Cf. A000010, A000225, A002326, A053287.

Magma
```
{* Order(g) : g in GF(2^6) | g ne 0 *};
```

f:= proc(n,k) if 2^n-1 mod k = 0 then numtheory:-phi(k) else 0 fi end proc:
seq(seq(f(n,k),k=1..2^n-1), n=1..10); # Robert Israel, Jul 21 2016

T[n_, k_] := If[Divisible[2^n - 1, k], EulerPhi[k], 0];
Table[T[n, k], {n, 1, 10}, {k, 1, 2^n - 1}] // Flatten (* Jean-François Alcover, Feb 07 2023, after Robert Israel *)

From Robert Israel, Jul 21 2016: (Start)
T(n,k) = A000010(k) if k is a divisor of 2^n-1, otherwise 0.
Sum_{k=1..2^n-1} T(n,k) = 2^n-1 = A000225(n).
G.f. as triangle: g(x,y) = Sum_{j>=0} x^A002326(j)*A000010(2j+1)*y^(2j+1)/(1-x^A002326(j)). (End)

T(6,21) corrected by Robert Israel, Jul 21 2016

A092589 a(n) = -A065395(2^n).

Original entry on oeis.org

0, 1, 3, 1, 15, 5, 63, 1, 177, 89, 913, -319, 4095, 2393, 10617, 1, 65535, 8897, 262143, -44287, 729537, 543553, 4015777, -1753087, 15622785, 11162969, 46358529, -1452031, 265390977, -2270911, 1073741823, 1, 2668569153, 2862962009, 15344762817, -8238350335, 68103158337, 45811586393

Offset: 0

Labos Elemer, Mar 02 2004

sign

Amiram Eldar, Table of n, a(n) for n = 0..1205

Cf. A000010, A000203, A000225, A033632, A053287, A065395, A092584, A092585, A092586, A092587, A092588.

fs[x_] := EulerPhi[DivisorSigma[1, x]]; sf[x_] := DivisorSigma[1, EulerPhi[x]]; Table[fs[2^w]-sf[2^w], {w, 0, 65}]

a(n) = phi(2^(n+1)-1) - 2^n + 1 = A053287(n+1) - A000225(n). - Amiram Eldar, Jun 09 2024

Offset changed to 0, a(0) prepended and name corrected by Amiram Eldar, Jun 09 2024

A056742 a(n) = phi(2^n - 1)/2.

Original entry on oeis.org

1, 3, 4, 15, 18, 63, 64, 216, 300, 968, 864, 4095, 5292, 13500, 16384, 65535, 69984, 262143, 240000, 889056, 1320352, 4105040, 3317760, 16200000, 22358700, 56733696, 66382848, 266913216, 267300000, 1073741823, 1073741824

Offset: 2

Robert G. Wilson v, Aug 14 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 2..1206

Cf. A000010, A000225, A011260, A053287.

Table[EulerPhi[(2^n - 1)]/2, {n, 2, 40}]

a(n) = eulerphi(2^n - 1)/2; \\ Amiram Eldar, Jun 09 2024

a(n) = A000010(A000225(n))/2 = A053287(n)/2. - Amiram Eldar, Jun 09 2024

A163368 a(n) = phi(sigma(tau(n))).

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 6, 2, 6, 2, 4, 2, 6, 6, 2, 2, 4, 2, 4, 6, 6, 2, 8, 2, 6, 6, 4, 2, 8, 2, 4, 6, 6, 6, 12, 2, 6, 6, 8, 2, 8, 2, 4, 4, 6, 2, 6, 2, 4, 6, 4, 2, 8, 6, 8, 6, 6, 2, 12, 2, 6, 4, 4, 6, 8, 2, 4, 6, 8, 2, 12, 2, 6, 4, 4, 6, 8, 2, 6, 2, 6, 2, 12, 6

Offset: 1

Jaroslav Krizek, Jul 25 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000040, A000203, A006881, A053287, A062069, A062401, A120944.

[EulerPhi(SumOfDivisors(NumberOfDivisors(n))): n in [1..80]]; // Vincenzo Librandi, Dec 21 2016

with(numtheory): A163368:=n->phi(sigma(tau(n))): seq(A163368(n), n=1..150); # Wesley Ivan Hurt, Dec 19 2016

Table[EulerPhi[DivisorSigma[1, DivisorSigma[0, n]]], {n, 100}] (* G. C. Greubel, Dec 19 2016 *)

vector(50, n, eulerphi(sigma(numdiv(n)))) \\ G. C. Greubel, Dec 19 2016

a(n) = A000010(A000203(A000005(n))) = A000010(A062069(n)) = A062401(A000005(n)).

a(1) = 1, a(p) = 2 for p = primes (A000040), a(pq) = 6 for pq = product of two distinct primes (A006881), a(pq...z) = A000010(2^(k+1)-1) = A053287(k+1) for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944).

A323616 a(n) is the largest prime factor of phi(2^n-1), where phi is Euler's totient.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 257, 3, 73, 5, 7, 31, 97, 5, 5, 13, 19, 7, 29, 11, 331, 2, 293, 257, 439, 3, 1039, 73, 389, 257, 8501, 43, 2713, 31, 37, 683, 569, 7, 5419, 5, 257, 31, 131, 19, 29, 241, 10639, 2179, 8060489, 11, 1321, 331, 1289, 17449

Offset: 1

Jianing Song, Jan 20 2019

nonn

If a(n) <= 2, then a regular (2^n-1)-gon can be constructed using a straightedge and compass; if a(n) <= 3, then a regular (2^n-1)-gon can be constructed using a straightedge, compass and an angle-trisector, etc.
It appears that each value occurs only a few times (see the Example section below), but to prove this seems nearly impossible.
Although a(n) = gpf(n) for the first few n, it should more often be the case that a(n) is relatively large compared to n. It seems that gpf(phi(2^n-1)) = gpf(n) only for n = 1..16, 18, 20, 21, 25, 26, 28, 29, 32, 36, 50. See the Example section below.
Nevertheless, there are many n such that a(n) = a(2*n) (including 44 of the first 100 terms). Moreover, if phi(2^n-1) and phi(2^n+1) have exactly the same prime factors, then phi(2^(2*n)-1) = phi(2^n-1)*phi(2^n+1) is powerful, and this holds for 2*n = 4, 6, 8, 12, 14, 16, 18, 26, 32, 36, 38, 50, 60, 62, 76, 108, 122, 254. By the way, phi(2^n-1) is also powerful for n = 9, 11, 15, 21, 25, 28, and there seem to be no other such numbers n.

			In the following list, a number k such that gpf(phi(2^k-1)) = gpf(k) is denoted with a "*".
a(n) = 1: 1* (1)
a(n) = 2: 2*, 4*, 8*, 16*, 32* (5)
a(n) = 3: 3*, 6*, 9*, 12*, 18*, 36* (6)
a(n) = 5: 5*, 10*, 15*, 20*, 24, 25*, 50* (7)
a(n) = 7: 7*, 14*, 21*, 28*, 48 (5)
a(n) = 11: 11*, 30, 60 (3)
a(n) = 13: 13*, 26* (2)
a(n) = 17: (0)
a(n) = 19: 27, 54, 108 (3)
a(n) = 23: (0)
a(n) = 29: 29*, 55 (2)
a(n) = 31: 22, 44, 52 (3)

Amiram Eldar, Table of n, a(n) for n = 1..512 (terms 1..250 from Jianing Song)

Cf. A000010, A006530, A053287.

a[n_] := FactorInteger[EulerPhi[2^n-1]][[-1, 1]]; Array[a, 64] (* Amiram Eldar, Mar 02 2025 *)

gpf(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
a(n) = gpf(eulerphi(2^n-1))

a(n) = A006530(A053287(n)).

cp's OEIS Frontend

A295503 a(n) = phi(10^n-1), where phi is Euler's totient function (A000010).

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A096853 a(n) = A062401(2^n-1).

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A096854 a(n) = A062402(2^n-1).

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A096855 a(n) = A062401(2^n + 1).

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A096856 a(n) = A062402(2^n+1).

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A057764 Triangle T(n,k) = number of nonzero elements of multiplicative order k in Galois field GF(2^n) (n >= 1, 1 <= k <= 2^n-1).

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A092589 a(n) = -A065395(2^n).

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A056742 a(n) = phi(2^n - 1)/2.

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