A020491 -id:A020491 - cp's OEIS frontend

A076688 a(n) = 2^(2^n-n-2).

1, 8, 1024, 33554432, 72057594037927936, 664613997892457936451903530140172288, 113078212145816597093331040047546785012958969400039613319782796882727665664

Offset: 2

Benoit Cloitre, Oct 25 2002

nonn

Integer values of 1/(2-Sum_{i=1..m} i/2^i).

The next term a(9) has 151 digits, and is too large to include in data. - Bernard Schott, Aug 27 2020

Cf. A036799.
Subsequence of A289585.
Cf. A000225, A020491, A058891.

Data := [seq(2^(2^n-n-2),  n = 2..8)]; \\ Bernard Schott, Aug 26 2020

Table[2^(2^n - n - 2), {n, 2, 8}] (* Amiram Eldar, Aug 27 2020 *)

A341939 Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Original entry on oeis.org

1, 3, 8, 10, 18, 19, 24, 30, 34, 45, 52, 57, 73, 74, 85, 102, 125, 135, 140, 152, 153, 156, 163, 182, 185, 190, 202, 219, 222, 252, 255, 333, 342, 360, 375, 394, 416, 420, 436, 451, 455, 456, 459, 476, 489, 505, 514, 546, 555, 570, 584, 606, 625, 629, 640, 646, 679, 680, 730

Offset: 1

Bernard Schott, Feb 24 2021

nonn

The first 11 terms of this sequence are also the first 11 terms of A341938: m such that phi(m)*tau(m) is a square, then, a(12) = 57 while A341938(12) = 54. Indeed, if phi(m)/tau(m) is a perfect square then phi(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A341940, the first one is 54 (last example).
Some subsequences (see examples):
-> The seven terms that satisfy also tau(m) = phi(m) form the subsequence A020488 with phi(m)/tau(m) = 1^2.
-> Primes p of the form 2*k^2 + 1 (A090698) form another subsequence because tau(p) = 2 and phi(p) = p-1 = 2*k^2, so phi(p)/tau(p) = k^2.
-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset because if p = 2, phi(8)/tau(8)=1, and if p odd, phi(p^3)/tau(p^3) = (k*p/2)^2 with k even.

			phi(30) = 8, tau(30) = 8 so phi(30)/tau(30) = 1^2, and 30 is a term.
phi(45) = 24, tau(45) = 6, so phi(45)/tau(45) = 4 = 2^2, and 85 is a term.
phi(125) = 100, tau(125) = 4, so phi(125)/tau(125) = 25 = 5^2, and 125 is a term.
phi(54) = 18, tau(54) = 8, and phi(54)/tau(54) = 18/8 = 9/4 = (3/2)^2 and 54 is not a term while phi(54)*tau(54) = 12^2.

Intersection of A020491 and A341938.
Similar for: A144695 (sigma(n)/tau(n) perfect square), A293391 (sigma(n)/phi(n) perfect square).
Subsequences: A090698, A020488.
Cf. A069237, A289585, A341940.
Cf. A000005 (phi), A000010(tau).

with(numtheory): filter:= q -> phi(q)/tau(q) = floor(phi(q)/tau(q)) and issqr(phi(q)/tau(q)) : select(filter, [$1..750]);

Select[Range[1000], IntegerQ @ Sqrt[EulerPhi[#]/DivisorSigma[0, #]] &] (* Amiram Eldar, Feb 24 2021 *)

isok(m) = my(x=eulerphi(m)/numdiv(m)); (denominator(x)==1) && issquare(x); \\ Michel Marcus, Feb 24 2021

A342665 Numbers k for which phi(k)+1 is a multiple of d(k), where phi is Euler totient function (A000010) and d(n) gives the number of divisors of n (A000005).

Original entry on oeis.org

1, 2, 4, 25, 81, 121, 289, 529, 841, 1681, 2209, 2809, 3481, 5041, 6889, 7921, 10201, 11449, 12100, 12769, 17161, 18769, 22201, 27889, 28561, 28900, 29929, 32041, 36481, 38809, 51529, 54289, 57121, 63001, 66049, 69169, 72361, 78961, 84100, 85849, 96721, 100489, 120409, 124609, 128881, 146689, 151321, 160801, 175561

Offset: 1

Antti Karttunen, Mar 30 2021

nonn

Numbers k such that A124331(k) = k. This is also a subsequence of the records of A124331 (both their values and their positions).

Terms other than 2 are a perfect square. Proof: phi(k) is even for k > 2. So phi(k)+1 is odd for k > 2. d(k) is odd only if k is a perfect square. So for any term k > 2 we need k to be a perfect square. Checking cases <= 2 leaves only 2 as the nonsquare in this sequence. - David A. Corneth, Mar 31 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A020491.

Fixed points of A124331. After 1, a subsequence of A015733.

Select[Join[{1, 2}, Range[2, 420]^2], Divisible[EulerPhi[#] + 1, DivisorSigma[0, #]] &] (* Amiram Eldar, Mar 31 2021 *)

isA342665(n) = !((eulerphi(n)+1) % numdiv(n));

A015732 Even numbers k such that d(k) | phi(k).

Original entry on oeis.org

8, 10, 18, 24, 26, 28, 30, 34, 40, 52, 56, 58, 70, 72, 74, 76, 78, 82, 84, 88, 90, 98, 102, 104, 106, 108, 110, 120, 122, 124, 126, 128, 130, 136, 140, 146, 148, 152, 156, 168, 170, 172, 174, 176, 178, 182, 184, 190, 194, 198, 202, 210, 218, 222, 224, 226, 228

Offset: 1

Robert G. Wilson v

nonn

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

Subsequence of A020491.

Cf. A000005 (d), A000010 (phi).

Select[Range[0, 250, 2], Divisible[EulerPhi[#], DivisorSigma[0, #]] &] (* Amiram Eldar, Dec 05 2024 *)

is(k) = if(k % 2, 0, my(f = factor(k)); !(eulerphi(f) % numdiv(f))); \\ Amiram Eldar, Dec 05 2024

A224108 Numbers k such that tau(k) divides k, sigma(k) and phi(k).

Original entry on oeis.org

1, 56, 184, 248, 376, 504, 568, 632, 672, 824, 864, 1016, 1208, 1248, 1336, 1528, 1592, 1656, 1784, 1824, 1912, 2016, 2104, 2168, 2232, 2488, 2688, 2872, 2936, 2976, 3064, 3360, 3384, 3448, 3512, 3552, 3704, 3832, 3896, 3968, 4024, 4128, 4284, 4320, 4792, 4856, 5048

Offset: 1

Jayanta Basu, Mar 31 2013

nonn

4 | a(n) for n > 1. Natural density 0. - Charles R Greathouse IV, Mar 31 2013

Zelinsky (2002) called these terms "rare numbers", and noted that if p and q are distinct primes, not equal to 2,3 or 7, then 672*p*q is a term. He proved that for any k > 0 and for sufficiently large m the number of terms not exceeding m is > k*pi(m), where pi(m) = A000720(m). - Amiram Eldar, Feb 20 2021

			56 is in the sequence because 56 has 8 divisors (1, 2, 4, 7, 8, 14, 28, 56), and 8 is a divisor of 56, as well as of sigma(56) = 120 and of phi(56) = 24.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Intersection of A003601, A020491, and A033950.

Cf. A000005, A000010, A000203, A000720, A069810, A217301.

Select[Range[10000], GCD[DivisorSigma[1, #], #, EulerPhi[#], DivisorSigma[0, #]] == DivisorSigma[0, #] &]
Select[Range[5100],AllTrue[{#,DivisorSigma[1,#],EulerPhi[#]}/ DivisorSigma[ 0,#], IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 02 2019 *)

is(n)=my(t=numdiv(n)); n%t==0 && sigma(n)%t==0 && eulerphi(n)%t==0 \\ Charles R Greathouse IV, Mar 31 2013

A290634 Positive integers which are never the quotient of phi(n)/tau(n).

Original entry on oeis.org

17, 19, 31, 38, 47, 59, 61, 62, 71, 85, 91, 101, 103, 107, 109, 118, 121, 133, 137, 149, 151, 157, 167, 181, 187, 197, 211, 217, 218, 223, 227, 229, 241, 247, 257, 259, 263, 266, 269, 271, 283, 289, 305, 311, 313, 314, 317, 327, 331, 334, 337, 347, 349, 353, 355, 361, 367

Offset: 1

Bernard Schott, Aug 08 2017

nonn

For phi(n)/tau(n) see A279287/A279288.
Numbers that do not appear in A175667.
The first nine terms of this sequence are exactly A119480(3) through A119480(11), and many other terms are common to these two sequences.

Cf. A000010, A000005, A020491, A015733, A020488, A062516, A175667, A112954, A112955, A119480, A289585, A279287/A279288.

A358061 a(n) = phi(n) mod tau(n).

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 3, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 6, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, 0, 4, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 1

Ctibor O. Zizka, Oct 28 2022

nonn

a(n) > 0 for n in A015733, a(n) = 0 for n in A020491.

			For n = 4; a(4) = A000010(4) mod A000005(4) = 2 mod 3 = 2.

Cf. A000005 (tau), A000010 (phi), A015733, A020491.

a[n_] := Mod[EulerPhi[n], DivisorSigma[0, n]]; Array[a, 100] (* Amiram Eldar, Oct 28 2022 *)

from math import prod
from sympy import factorint
def A358061(n):
    f = factorint(n).items()
    d = prod(e+1 for p, e in f)
    return prod(pow(p,e-1,d)*((p-1)%d) for p, e in f) % d # Chai Wah Wu, Oct 29 2022

a(n) = A000010(n) mod A000005(n).

A378089 Irregular triangle read by rows in which row n lists the numbers k such that phi(k)/tau(k) = n.

Original entry on oeis.org

1, 3, 8, 10, 18, 24, 30, 5, 9, 15, 28, 40, 72, 84, 90, 120, 7, 21, 26, 56, 70, 78, 108, 126, 168, 210, 34, 45, 52, 102, 140, 156, 252, 360, 420, 11, 33, 88, 110, 198, 264, 330, 13, 35, 39, 63, 76, 104, 105, 130, 228, 234, 280, 312, 390, 504, 540, 630, 840, 58, 98, 174, 294

Offset: 1

Mohammed Yaseen, Nov 16 2024

			Triangle begins:
  n=1: 1, 3, 8, 10, 18, 24, 30;
  n=2: 5, 9, 15, 28, 40, 72, 84, 90, 120;
  n=3: 7, 21, 26, 56, 70, 78, 108, 126, 168, 210;
  n=4: 34, 45, 52, 102, 140, 156, 252, 360, 420;
  n=5: 11, 33, 88, 110, 198, 264, 330;
  n=6: 13, 35, 39, 63, 76, 104, 105, 130, 228, 234, 280, 312, 390, 504, 540, 630, 840;
  n=7: 58, 98, 174, 294;
  ...

Cf. A000005 (tau), A000010 (phi).
Cf. A020488 (row 1), A062516 (row 2), A063469 (row 3), A063470 (row 4).
Cf. A112954 (row lengths), A175667 (1st column), A112955 (right column), A020491 (ordered terms).

cp's OEIS Frontend

A076688 a(n) = 2^(2^n-n-2).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

A341939 Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

A342665 Numbers k for which phi(k)+1 is a multiple of d(k), where phi is Euler totient function (A000010) and d(n) gives the number of divisors of n (A000005).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A015732 Even numbers k such that d(k) | phi(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A224108 Numbers k such that tau(k) divides k, sigma(k) and phi(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A290634 Positive integers which are never the quotient of phi(n)/tau(n).

Views

Author

Keywords

Comments

Crossrefs

A358061 a(n) = phi(n) mod tau(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Formula

A378089 Irregular triangle read by rows in which row n lists the numbers k such that phi(k)/tau(k) = n.

Views

Author

Keywords

Examples

Crossrefs