A032447 -id:A032447 - cp's OEIS frontend

A035114 Values of phi(n) corresponding to A035113.

1, 2, 2, 4, 4, 4, 6, 6, 8, 8, 8, 8, 10, 12, 12, 12, 12, 16, 16, 16, 16, 16, 18, 18, 20, 20, 20, 22, 24, 24, 24, 24, 24, 24, 24, 28, 30, 32, 32, 32, 32, 32, 32, 36, 36, 36, 36, 36, 40, 40, 40, 40, 40, 40, 42, 42, 44, 44, 46, 48, 48, 48, 48, 48, 48, 48, 48, 48

Offset: 1

N. J. A. Sloane

			phi(1)=1, phi(3)=2, phi(4)=2, phi(5)=4, ...

Dumitru Damian, Table of n, a(n) for n = 1..10000
L. C. Washington, The Main Conjecture and Annihilation of Class Groups, In: Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol 83 (1997) Springer, New York, NY.

Cf. A000010, A035113.

Cf. A002181, A002202, A007614, A014197, A016825, A032447, A058277.

from sympy import totient as A000010
def lov(n): return sorted([[A000010(n), n] for n in range(1,n) if n%4 != 2])
print([x[0] for x in lov(200)][:100]) # Dumitru Damian, Feb 03 2022

a(n) = A000010(A035113(n)). - Michel Marcus, Feb 07 2022

More terms from James Sellers

a(43) onward corrected by Sean A. Irvine, Sep 26 2020

A165773 Numbers n for which phi(n) = m! for some integer m, where phi = A000010.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 14, 18, 35, 39, 45, 52, 56, 70, 72, 78, 84, 90, 143, 155, 175, 183, 225, 231, 244, 248, 286, 308, 310, 350, 366, 372, 396, 450, 462, 779, 793, 803, 905, 925, 1001, 1045, 1085, 1107, 1209, 1221, 1281, 1287, 1395, 1425, 1448, 1485, 1558, 1575

Offset: 1

M. F. Hasler, Oct 02 2009

A subsequence of A032447. Can be read as "fuzzy" table, where the m-th row contains A055506(m) numbers with phi=m!, ranging from A055487(m) to A165774(m). See there for more information.

A log-log plot shows the components of this sequence better. - T. D. Noe, Jun 21 2012

			The table looks as follows:
1,2, /* A055506(1)=2 numbers for which phi(n) = 1! = 1 */
3,4,6, /* A055506(2)=3 numbers for which phi(n) = 2! = 2 */
7,9,14,18, /* A055506(3)=4 numbers for which phi(n) = 3! = 6 */
35,39,45,52,56,70,72,78,84,90, /* A055506(4)=10 numbers for which phi(n) = 4! = 24 */
143,155,175,183,225,231,244,248,286,308,310,350,366,372,396,450,462, /* A055506(5)=17 numbers for which phi(n) = 5! = 120 */ ...

T. D. Noe, Table of n, a(n) for n = 1..5477 (for factorials up to 10)
T. D. Noe, A log-log plot

for(m=1,8, for( n=f=m!,f*(m+1), eulerphi(n)==f & print1(n","));print())

Fixed references to A055506, A055487 and A165774 - M. F. Hasler, Oct 04 2009

A215240 Sum of the numbers p such that phi(p) = n, where phi is Euler's totient function.

Original entry on oeis.org

3, 13, 0, 35, 0, 48, 0, 105, 0, 33, 0, 166, 0, 0, 0, 231, 0, 138, 0, 218, 0, 69, 0, 621, 0, 0, 0, 87, 0, 93, 0, 581, 0, 0, 0, 655, 0, 0, 0, 833, 0, 276, 0, 299, 0, 141, 0, 1514, 0, 0, 0, 159, 0, 243, 0, 377, 0, 177, 0, 1114, 0, 0, 0, 1315, 0, 201, 0, 0, 0, 213, 0

Offset: 1

T. D. Noe, Oct 12 2012

nonn

These terms (greater than 0) are not unique. The first duplicate appears at a(256) = a(2236) = 6711.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
David M. Bressoud, A Course in Computational Number Theory (web page), CNT.m, Computational Number Theory Mathematica package.

Cf. A002181 (smallest inverse), A006511 (largest inverse), A217842 (product of inverses).

Cf. A007617, A032447 (inverse of phi).

Needs["CNT`"]; Table[Total[PhiInverse[n]], {n, 100}]

a(n) = vecsum(invphi(n)); \\ Amiram Eldar, Nov 15 2024, using Max Alekseyev's invphi.gp

a(n) = 0 if and only if n is in A007617. - Amiram Eldar, Nov 15 2024

A231612 Numbers n such that the four fourth-degree cyclotomic polynomials are simultaneously prime.

Original entry on oeis.org

2, 90750, 194468, 229592, 388332, 868592, 1054868, 1148390, 1380380, 1415920, 1461372, 1496010, 1614800, 1706398, 1992210, 2439042, 2478212, 2644498, 2791910, 3073300, 3264448, 3824370, 3892780, 3939222, 3941938, 4425970, 4468980, 4594138, 4683700

Offset: 1

T. D. Noe, Dec 11 2013

nonn

The polynomials are cyclotomic(5,x) = 1 + x + x^2 + x^3 + x^4, cyclotomic(8,x) = 1 + x^4, cyclotomic(10,x) = 1 - x + x^2 - x^3 + x^4, and cyclotomic(12,x) = 1 - x^2 + x^4. The numbers 5, 8, 10, and 12 are in the fourth row of A032447.

By Schinzel's hypothesis H, there are an infinite number of n that yield simultaneous primes. Note that the two first-degree cyclotomic polynomials, x-1 and x+1, yield the twin primes for the numbers in A014574.

See A087277.

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).
Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials).

Select[Range[5000000], PrimeQ[Cyclotomic[5, #]] && PrimeQ[Cyclotomic[8, #]] && PrimeQ[Cyclotomic[10, #]] && PrimeQ[Cyclotomic[12, #]] &]
Select[Range[47*10^5],AllTrue[Thread[Cyclotomic[{5,8,10,12},#]],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 22 2018 *)

A231613 Numbers n such that the four sixth-degree cyclotomic polynomials are simultaneously prime.

Original entry on oeis.org

32034, 162006, 339105, 458811, 1780425, 2989119, 2993100, 3080205, 4375404, 6129597, 6280221, 7565142, 8489820, 10268277, 11343741, 12065076, 13067295, 13333182, 15866508, 16472802, 17040537, 18028605, 19066758, 22633629, 24256362, 24365259, 25031349

Offset: 1

T. D. Noe, Dec 11 2013

nonn

The polynomials are cyclotomic(7,x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6, cyclotomic(9,x) = 1 + x^3 + x^6, cyclotomic(14,x) = 1 - x + x^2 - x^3 + x^4 - x^5 + x^6, and cyclotomic(18,x) = 1 - x^3 + x^6. The numbers 7, 9, 14 and 18 are in the sixth row of A032447.

By Schinzel's hypothesis H, there are an infinite number of n that yield simultaneous primes. Note that the two first-degree cyclotomic polynomials, x-1 and x+1, yield the twin primes for the numbers in A014574.

See A087277.

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials).
Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials).

t = {}; n = 0; While[Length[t] < 30, n++; If[PrimeQ[Cyclotomic[7, n]] && PrimeQ[Cyclotomic[9, n]] && PrimeQ[Cyclotomic[14, n]] && PrimeQ[Cyclotomic[18, n]], AppendTo[t, n]]]; t
Select[Range[251*10^5],AllTrue[Cyclotomic[{7,9,14,18},#],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 29 2016 *)

A231614 Numbers n such that the five eighth-degree cyclotomic polynomials are simultaneously prime.

Original entry on oeis.org

4069124, 8919014, 8942756, 46503870, 75151624, 82805744, 189326670, 197155324, 271490544, 365746304, 648120564, 1031944990

Offset: 1

T. D. Noe, Dec 11 2013

The polynomials are cyclotomic(15,x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8, cyclotomic(16,x) = 1 + x^8, cyclotomic(20,x) = 1 - x^2 + x^4 - x^6 + x^8, cyclotomic(24,x) = 1 - x^4 + x^8, and cyclotomic(30,x) = 1 + x - x^3 - x^4 - x^5 + x^7 + x^8. The numbers 15, 16, 20, 24 and 30 are in the eighth row of A032447.

By Schinzel's hypothesis H, there are an infinite number of n that yield simultaneous primes. Note that the two first-degree cyclotomic polynomials, x-1 and x+1, yield the twin primes for the numbers in A014574.

See A087277.

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials).
Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).

t = {}; n = 0; While[Length[t] < 6, n++; If[PrimeQ[Cyclotomic[15, n]] && PrimeQ[Cyclotomic[16, n]] && PrimeQ[Cyclotomic[20, n]] && PrimeQ[Cyclotomic[24, n]] && PrimeQ[Cyclotomic[30, n]], AppendTo[t, n]]]; t

Extended to 12 terms by T. D. Noe, Dec 13 2013

A289276 Numbers k such that phi(k) (the totient function A000010) is a power of the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 17, 18, 24, 30, 34, 63, 76, 85, 128, 136, 170, 257, 315, 333, 364, 380, 436, 444, 514, 640, 680, 972, 1285, 1542, 1820, 1824, 1836, 1875, 2142, 2220, 2907, 3285, 3488, 3796, 4369, 4788, 4860

Offset: 1

Franklin T. Adams-Watters, Jun 30 2017

A019434 is a subsequence. - David A. Corneth, Jun 30 2017

Is the frequency of e such that A000005(a(n))^e = A000010(a(n)) finite? - David A. Corneth, Jul 01 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..5319 (first 526 terms from Antti Karttunen)

Cf. A000005, A000010, A019434, A020488, A032447, A036913, A051281, A068559, A068560, A114063, A286627.

Join[{1},Select[Range[2,5000],IntegerQ[Log[DivisorSigma[0,#],EulerPhi[#]]]&]] (* Harvey P. Dale, Aug 06 2017 *)

ispowerof(n, k)= if(k==1, return(n==1)); while(n>=k, if(n%k!=0, return(0)); n\=k); n==1
isa(n) = ispowerof(eulerphi(n),numdiv(n)) \\ Quick program, fast enough for early values.

is(n) = if(n==1, return(1)); my(f = factor(n); phi = eulerphi(f), ndiv = numdiv(f), e = logint(phi, ndiv)); ndiv^e == phi \\ David A. Corneth, Jun 30 2017, changed per suggestion of Charles R Greathouse IV

isA289276(n)= if(n==1, return(1)); my(phi = eulerphi(n), ndiv = numdiv(n), v = valuation(phi, ndiv)); ndiv^v == phi; \\ (A variant of above program). - Antti Karttunen, Jun 30 2017

list(lim)=my(v=List([1])); forfactored(n=2,lim\1, my(phi = eulerphi(n), ndiv = numdiv(n)); if(ndiv^valuation(phi,ndiv) == phi, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 01 2017

A362180 Irregular table read by rows in which the n-th row consists of all the numbers m such that A323410(m) = n.

Original entry on oeis.org

6, 10, 12, 15, 14, 20, 21, 18, 24, 28, 35, 22, 36, 40, 33, 45, 26, 44, 56, 39, 55, 63, 52, 72, 65, 77, 34, 48, 88, 51, 91, 99, 38, 68, 80, 104, 57, 85, 117, 30, 76, 112, 95, 119, 143, 46, 136, 144, 69, 133, 153, 50, 92, 152, 176, 75, 115, 171, 187, 54, 100, 208

Offset: 2

Amiram Eldar, Apr 10 2023

The offset is 2 since A323410(p) = 1 for all prime powers p (A246655).

The 0th row consists of one term, 1, since 1 is the only solution to A323410(x) = 0.

			The table begins:
  n   n-th row
  --  -----------
   2
   3
   4  6;
   5
   6  10, 12;
   7  15;
   8  14, 20;
   9  21;
  10  18, 24, 28;
  11  35;
  12  22, 36, 40;

Amiram Eldar, Table of n, a(n) for n = 2..12684 (rows 2..1000)

Cf. A246655, A323410, A362181 (row lengths).

Similar sequences: A032447, A361966, A362213.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0;
With[{max = 28}, ucot = Table[ucototient[n], {n, 1, max^2}]; row[n_] := Position[ucot, n] // Flatten; Table[row[n], {n, 2, max}] // Flatten]

A293928 Totients phi(m) having one or more solutions m to phi(m)^2 = phi(phi(m)*m).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 84, 96, 100, 108, 120, 128, 144, 160, 162, 168, 192, 200, 216, 240, 252, 256, 272, 288, 312, 320, 324, 336, 360, 384, 400, 432, 440, 480, 486, 500, 504, 512, 544, 576, 588, 600, 624, 640, 648, 672, 684

Offset: 1

Torlach Rush, Oct 19 2017

nonn

"Totients" are terms of A000010. - N. J. A. Sloane, Oct 22 2017
The smallest totient absent from the list is 10. This is because the totient inverses of 10, 11 and 22 are not solutions to phi(m)^2 = phi(phi(m)*m).
The formula is recursive. For example, taking a(22) we get the following: 11664 = phi(108*324), 1259712 = phi(11664*324), 136048896 = phi(1259712*324), ...
Where (if ever) does this first differ from A068997? - R. J. Mathar, Oct 30 2017
Apparently the set of the m is A151999. - R. J. Mathar, Mar 25 2024
If m satisfies phi(m)^2 = phi(phi(m)*m), then it satisfies phi(m)^(k+1) = phi(phi(m)^k*m) for all k >= 1. - Max Alekseyev, Dec 03 2024

			96 is a term since 96^2 = phi(96*288), with m=288 where phi(288) = 96.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A006511, A032447, A007366.

Subsequence of A002202.

isok(n) = {my(iv = invphi(n)); if (#iv, for (m = 1, #iv, if (n^2 == eulerphi(n*iv[m]), return (1)););); return (0);} \\ using the invphi script by Max Alekseyev; Michel Marcus, Nov 01 2017

More terms from Michel Marcus, Oct 24 2017

Definition simplified by Max Alekseyev, Dec 03 2024

A362213 Irregular table read by rows in which the n-th row consists of all the numbers m such that cototient(m) = n, where cototient is A051953.

Original entry on oeis.org

4, 9, 6, 8, 25, 10, 15, 49, 12, 14, 16, 21, 27, 35, 121, 18, 20, 22, 33, 169, 26, 39, 55, 24, 28, 32, 65, 77, 289, 34, 51, 91, 361, 38, 45, 57, 85, 30, 95, 119, 143, 529, 36, 40, 44, 46, 69, 125, 133, 63, 81, 115, 187, 52, 161, 209, 221, 841, 42, 50, 58, 87, 247, 961

Offset: 2

Amiram Eldar, Apr 11 2023

The offset is 2 since cototient(p) = 1 for all primes p.

The 0th row consists of one term, 1, since 1 is the only solution to cototient(x) = 0.

			The table begins:
  n   n-th row
  --  -----------
   2  4;
   3  9;
   4  6, 8;
   5  25;
   6  10;
   7  15, 49;
   8  12, 14, 16;
   9  21, 27;
  10
  11  35, 121;
  12  18, 20, 22;

Amiram Eldar, Table of n, a(n) for n = 2..9674 (rows 2..1000)

Cf. A005278, A051953, A063507, A063740 (row lengths), A063741, A063742, A063748, A100827, A101373, A131825, A131826.

Similar sequences: A032447, A361966, A362180.

With[{max = 50}, cot = Table[n - EulerPhi[n], {n, 1, max^2}]; row[n_] := Position[cot, n] // Flatten; Table[row[n], {n, 2, max}] // Flatten]

cp's OEIS Frontend

A035114 Values of phi(n) corresponding to A035113.

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A165773 Numbers n for which phi(n) = m! for some integer m, where phi = A000010.

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A215240 Sum of the numbers p such that phi(p) = n, where phi is Euler's totient function.

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A231612 Numbers n such that the four fourth-degree cyclotomic polynomials are simultaneously prime.

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A231613 Numbers n such that the four sixth-degree cyclotomic polynomials are simultaneously prime.

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Mathematica

A231614 Numbers n such that the five eighth-degree cyclotomic polynomials are simultaneously prime.

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Mathematica

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A289276 Numbers k such that phi(k) (the totient function A000010) is a power of the number of divisors of k (A000005).

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A362180 Irregular table read by rows in which the n-th row consists of all the numbers m such that A323410(m) = n.

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