A055487 -id:A055487 - cp's OEIS frontend

A055506 Number of solutions to the equation phi(x) = n!.

2, 3, 4, 10, 17, 49, 93, 359, 1138, 3802, 12124, 52844, 182752, 696647, 2852886, 16423633, 75301815, 367900714, 1531612895, 8389371542, 40423852287, 213232272280, 1295095864798, 7991762413764, 42259876674716, 252869570952706, 1378634826630301, 8749244047999717

Offset: 1

Labos Elemer, Jun 29 2000

nonn

Note that if phi(x) = n!, then x must be a product of primes p such that p - 1 divides n!. - David Wasserman, Apr 30 2002

Gives the row lengths of the table A165773 (see example). All solutions to phi(x)=n! are in the interval [n!,(n+1)!] with the smallest/largest solutions given in A055487/A165774 respectively. - M. F. Hasler, Oct 04 2009

			n = 5, phi(x) = 5! = 120 holds for the following 17 numbers: { 143, 155, 175, 183, 225, 231, 244, 248, 286, 308, 310, 350, 366, 372, 396, 450, 462 }.
From _M. F. Hasler_, Oct 04 2009: (Start)
The table A165773 looks as follows:
  1,2, (a(1)=2 numbers for which phi(n) = 1! = 1)
  3,4,6, (a(2)=3 numbers for which phi(n) = 2! = 2)
  7,9,14,18, (a(3)=4 numbers for which phi(n) = 3! = 6)
  35,39,45,52,56,70,72,78,84,90, (a(4)=10 numbers for which phi(n) = 4! = 24)
  ... (End)

Andrew Lelechenko, Table of n, a(n) for n = 1..36
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A000142, A000203, A014197, A054873, A055486, A067847, A165773, A165774.

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

use ntheory ":all"; print "$ ",scalar(inverse_totient(factorial($))),"\n" for 1..20; # Dana Jacobsen, Mar 04 2019

a(n) = A014197(n!) = Cardinality({x; A000010(x) = A000142(n)}).

More terms from Jud McCranie, Jan 02 2001
More terms from David Wasserman, Apr 30 2002 (with the assistance of Vladeta Jovovic and Sascha Kurz)
a(21)-a(28) from Max Alekseyev, Jan 26 2012, Jul 09 2014

A055488 Smallest number x such that sum of divisors of x is n!.

Original entry on oeis.org

5, 14, 54, 264, 1560, 10920, 97440, 876960, 10263240, 112895640, 1348827480, 18029171160, 264370186080, 3806158356000, 62703141621120, 1128159304272000, 20422064875212000, 404757215566704000, 8208550091549808000, 177650747421074928000

Offset: 3

Labos Elemer, Jun 28 2000

nonn

For n = 1, a(1) = 1; for n = 2, there is no solution.

			For n = 6 the 15 solutions are as follows: {264,270,280,354,376,406,418,435,459,478,537,623,649,667,719}

R. K. Guy (1981): Unsolved Problems In Number Theory, B39.

Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2

Cf. A000203, A000142, A055486, A055487, A055489

a(n) = Min{x; Sigma[x] = n!} = Min{x; A000203(x) = A000142(n)}

More terms from Jud McCranie, Oct 09 2000
a(13), a(14) from Vim Wenders, Nov 06 2006, Jan 12 2007
a(15), a(16) from Donovan Johnson, Aug 26 2008, Mar 26 2010
a(17)-a(22) from Max Alekseyev, Jan 25 2012

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

A165774 Largest solution to phi(x) = n!, where phi() is Euler totient function (A000010).

Original entry on oeis.org

2, 6, 18, 90, 462, 3150, 22050, 210210, 1891890, 19969950, 219669450, 2847714870, 37020293310, 520843112790, 7959363061650, 135309172048050, 2300255924816850, 41996101027370490, 797925919520039310, 16504589035937252250, 347097774991217099850, 7751850308137181896650, 179602728970220622816750, 4493489228616853106091450, 112337230715421327652286250, 2958213742172761628176871250, 79871771038664563960775523750, 2279417465795734863803670716250

Offset: 1

M. F. Hasler, Oct 04 2009

nonn

All solutions to phi(x) = n! belong to the interval [n!,(n+1)!] and are listed in the n-th row of A165773 (when written as table with row lengths A055506). Thus this sequence gives the last element in these rows, i.e., a(n) = A165773(Sum_{k=1..n} A055506(k)).
All terms in this sequence are even, since if x is an odd solution to phi(x) = n!, then 2x is a larger solution because phi(2x) = phi(2)*phi(x) = phi(x).
Most terms (and any term divisible by 4) are divisible by 3, since if x = 2^k*y is a solution with k>1 and gcd(y,2*3) = 1, then x*3/2 = 2^(k-1)*3*y is a larger solution because phi(2^(k-1)*3) = 2^(k-2)*(3-1) = 2^(k-1) = phi(2^k).
For the same reason, most terms are divisible by 5, since if x=2^k*y is a solution with k>2 and gcd(y,2*5) = 1, then x*5/4 is a larger solution.
Also, any term of the form x = 2^k*3^m*y with k,m>1 must be divisible by 7 (else x*7/6 would be a larger solution), and so on.
Experimentally, a(n) = c(n)*(n+1)! with a coefficient c(n) ~ 2^(-n/10) (e.g., c(1) = c(2) = 1, c(10) ~ 0.5).

			a(1) = 2 is the largest among the A055506(1) = 2 solutions {1,2} to phi(n) = 1! = 1.
a(4) = 90 is the largest among the A055506(4) = 10 solutions {35, 39, 45, 52, 56, 70, 72, 78, 84, 90} to phi(n) = 4! = 24.
See A165773 for more examples.

Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A055487, A055506, A055489, A014197, A165773.

a(n) = invphiMax(n!); \\ Amiram Eldar, Nov 11 2024, using Max Alekseyev's invphi.gp

Edited and terms a(12)-a(28) added by Max Alekseyev, Jan 26 2012, Jul 09 2014

A123476 a(n) = (n!)^2/phi(n!), where phi is Euler's totient function.

Original entry on oeis.org

1, 4, 18, 72, 450, 2700, 22050, 176400, 1587600, 15876000, 192099600, 2305195200, 32464832400, 454507653600, 6817614804000, 109081836864000, 1970290678356000, 35465232210408000, 711274934886516000

Offset: 1

T. D. Noe, Sep 27 2006

nonn

a(n) is a solution to the equation phi(x) = n!.

T. D. Noe, Table of n, a(n) for n=1..100
P. Erdos and J. Lambek, Problem 4221, Amer. Math. Monthly, 55 (1948), 103.

Cf. A000010, A000142, A002202 (values taken by phi(x)).

Cf. A055487 (least m such that phi(m) = n!).

Table[(n!)^2/EulerPhi[n!],{n,20}] (* Harvey P. Dale, Nov 29 2015 *)

a(n) = (n!)^2/eulerphi(n!); \\ Michel Marcus, Jul 11 2018

A100013 Number of prime factors in n!+7 (counted with multiplicity).

Original entry on oeis.org

3, 3, 2, 1, 1, 1, 1, 3, 3, 3, 3, 2, 3, 3, 4, 2, 2, 3, 3, 5, 5, 5, 3, 4, 3, 2, 4, 5, 5, 4, 7, 6, 4, 4, 7, 2, 5, 4, 7, 4, 5, 3, 4, 6, 5, 4, 3, 3, 5, 6, 3, 5, 6, 3, 3, 7, 4, 5, 5, 2, 4, 4, 5, 4, 2, 4, 3, 5, 2, 5, 7, 4, 7, 5, 5, 3, 5, 4, 6, 6, 8, 5

Offset: 0

Jonathan Vos Post, Nov 18 2004

			Example 1!+7 = 2^3 so a(1) = 3.
a(3) = a(4) = a(5) = a(6) = 1 because 3!+1 = 13, 4!+7 = 31, 5!+1 = 127, 6!+7 = 727 and these are all primes. a(11) = a(15) = a(16) = a(25) = a(35) = a(59) = 2 because 11!+7 = 39916807 = 7 * 5702401, 15!+7 = 1307674368007 = 7 * 186810624001, 16!+7 = 20922789888007 = 7 * 2988969984001, 25!+7 = 15511210043330985984000007 = 7 * 2215887149047283712000001, 35!+7 = 10333147966386144929666651337523200000007 = 7 *
1476163995198020704238093048217600000001 and 59!+7 = 138683118545689835737939019720389406345902876772687432540821294940160000000000007 = 7 * 19811874077955690819705574245769915192271839538955347505831613562880000000000001 are all semiprimes.

C. Caldwell and H. Dubner, "Primorial, factorial and multifactorial primes," Math. Spectrum, 26:1 (1993/4) 1-7.

Chris Caldwell, multifactorial prime (another Prime Pages' Glossary entries).
Ken Davis, Status of Search for Multifactorial Primes.
Sean A. Irvine, Factorizations of n!+7

Cf. A002981, A037083, A037083, A085150, A085148, A085146, A062701, A055487, A062702.

More terms from Sean A. Irvine, Sep 20 2012

A066278 Least solution to cototient(x) = n!, where cototient(x) = x-phi(x).

Original entry on oeis.org

2, 4, 10, 36, 168, 1008, 7056, 56448, 470400, 4704000, 51528960, 618347520, 7740902400, 108372633600

Offset: 1

Labos Elemer, Dec 10 2001

a(15) <= 1625589504000. a(16) <= 26009432064000. a(17) <= 442160345088000. [Donovan Johnson, Feb 05 2010]

			Solutions to x-Phi[x]=720 are {1008,1424,1432,1436,1438} of which the smallest is a(6)=1008.

Cf. A000010, A051953, A000142, A055487.

a(n) = my(x=1); while(x-eulerphi(x) != n!, x++); x; \\ Michel Marcus, Aug 24 2019

a(n) = Min{x; x-A000010(x)=n!} = Min{A051953(x)=A000142(n)}.

a(12)-a(14) from Donovan Johnson, Feb 05 2010

A358489 Numbers k such that phi(k) = 13! where phi is the Euler totient function (A000010).

Original entry on oeis.org

6227180929, 6227182993, 6227186509, 6227199361, 6227220691, 6227229637, 6227245393, 6227246107, 6227260969, 6227267713, 6227268799, 6227279341, 6227280491, 6227288461, 6227311397, 6227314111, 6227327761, 6227351861, 6227355097, 6227376241, 6227447761, 6227454979

Offset: 1

Darío Clavijo, Nov 18 2022

A055487(13) = 6227180929 is the first term in the sequence.

A165774(13) = 37020293310 is the last term in the sequence.

Project Euler, Problem 248. Numbers for which Euler's totient function equals 13!.

Cf. A000010, A055487, A165774.

from sympy import totient
def isok(k): return totient(k) == 6227020800

cp's OEIS Frontend

A055506 Number of solutions to the equation phi(x) = n!.

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A055488 Smallest number x such that sum of divisors of x is n!.

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

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A165774 Largest solution to phi(x) = n!, where phi() is Euler totient function (A000010).

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A123476 a(n) = (n!)^2/phi(n!), where phi is Euler's totient function.

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Mathematica

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A100013 Number of prime factors in n!+7 (counted with multiplicity).

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A066278 Least solution to cototient(x) = n!, where cototient(x) = x-phi(x).

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A358489 Numbers k such that phi(k) = 13! where phi is the Euler totient function (A000010).

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