cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A055487 Least m such that phi(m) = n!.

Original entry on oeis.org

1, 3, 7, 35, 143, 779, 5183, 40723, 364087, 3632617, 39916801, 479045521, 6227180929, 87178882081, 1307676655073, 20922799053799, 355687465815361, 6402373865831809, 121645101106397521, 2432902011297772771, 51090942186005065121, 1124000727844660550281, 25852016739206547966721, 620448401734814833377121, 15511210043338862873694721, 403291461126645799820077057, 10888869450418352160768000001, 304888344611714964835479763201
Offset: 1

Views

Author

Labos Elemer, Jun 28 2000

Keywords

Comments

Erdős believed (see Guy reference) that phi(x) = n! is solvable.
Factorial primes of the form p = A002981(m)! + 1 = k! + 1 give the smallest solutions for some m (like m = 1,2,3,11) as follows: phi(p) = p-1 = A002981(m)!.
According to Tattersall, in 1950 H. Gupta showed that phi(x) = n! is always solvable. - Joseph L. Pe, Oct 01 2002
A123476(n) is a solution to the equation phi(x)=n!. - T. D. Noe, Sep 27 2006
From M. F. Hasler, Oct 04 2009: (Start)
Conjecture: Unless n!+1 is prime (i.e., n in A002981), a(n)=pq where p is the least prime > sqrt(n!) such that (p-1) | n! and q=n!/(p-1)+1 is prime.
Probably "least prime > sqrt(n!)" can also be replaced by "largest prime <= ceiling(sqrt(n!))". The case "= ceiling(...)" occurs for n=5, sqrt(120) = 10.95..., p=11, q=13.
a(n) is the first element in row n of the table A165773, which lists all solutions to phi(x)=n!. Thus a(n) = A165773((Sum_{kA055506(k)) + 1). The last element of each row (i.e., the largest solution to phi(x)=n!) is given in A165774. (End)

References

  • R. K. Guy, (1981): Unsolved problems In Number Theory, Springer - page 53.
  • Tattersall, J., "Elementary Number Theory in Nine Chapters", Cambridge University Press, 2001, p. 162.

Links

Crossrefs

Cf. A055486, A055488, A055489, A055506, A000010, A000142.
Cf. A123476, A165773, A165774.

Programs

  • Mathematica
    Array[Block[{k = 1}, While[EulerPhi[k] != #, k++]; k] &[#!] &, 10] (* Michael De Vlieger, Jul 12 2018 *)

Formula

a(n) = Min{m : phi(m) = n!} = Min{m : A000010(m) = A000142(n)}.

Extensions

More terms from Don Reble, Nov 05 2001
a(21)-a(28) from Max Alekseyev, Jul 09 2014

A055486 Number of solutions to sigma(x) = n!.

Original entry on oeis.org

1, 0, 1, 3, 4, 15, 33, 111, 382, 1195, 3366, 14077, 53265, 229603, 910254, 4524029, 18879944, 91336498, 561832582, 2801857644, 14652294729, 78894985156, 408373652461, 2378940665083, 11939275822636, 71931330299023, 392274481206066, 2626331088771946
Offset: 1

Views

Author

Labos Elemer, Jun 28 2000

Keywords

Examples

			For n = 9, solutions to sigma(x) = n! = 362880 form a set {97440, ..., 361657} of size 382, so a(9) = 382.

References

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

Links

Crossrefs

Cf. A000142, A000203, A054973, A014197, A055488, A055489, A055506.

Programs

  • Maple
    with(numtheory): for f from 1 to 9 do fac := f!: k := 0:for n from 1 to fac do if sigma(n)=fac then k := k+1: fi: od: print( k); od:

Formula

a(n) = A054973(n!) = Cardinality[{x; A000203(x) = A000142(n)}].

Extensions

More terms from Jud McCranie, Oct 09 2000
a(13)-a(14) from Donovan Johnson, Nov 22 2008
a(15) from Ray Chandler, Jan 13 2009
a(16)-a(28) from Max Alekseyev, Jan 23 2012

A055489 Largest number x such that sum of divisors of x is n!.

Original entry on oeis.org

5, 23, 95, 719, 5039, 39917, 361657, 3624941, 39904153, 479001599, 6226862869, 87178291199, 1307672080867, 20922780738961, 355687390376431, 6402373545694717, 121645099711277873, 2432902005056589697, 51090942157413850441
Offset: 3

Views

Author

Labos Elemer, Jun 28 2000

Keywords

Comments

For n = 1, a(1) = 1; for n = 2 there is no solution.
For n in A002982, a(n) = n!-1.

Examples

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

References

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

Links

Crossrefs

Cf. A000203, A000142, A002982, A055486, A055488, A057637.

Programs

  • PARI
    a(n) = {fn = n!; x = fn - 1; while(sigma(x) != fn, x--); x;} \\ Michel Marcus, Dec 17 2013

Formula

a(n) = Max{x; Sigma[x] = n!} = Max{x; A000203(x) = A000142(n)}
a(n) = A057637(A000142(n)). - Ray Chandler, Jan 15 2009
a(A002982(n)) = A000142(A002982(n)) - 1. - Ray Chandler, Jan 15 2009

Extensions

More terms from Jud McCranie, Oct 09 2000
a(15) from Donovan Johnson, Aug 31 2008
a(16)-a(19) from Donovan Johnson, Nov 22 2008
a(20)-a(52) from Ray Chandler, Jan 15 2009

A354074 Factorials that are the sum of the divisors of some m.

Original entry on oeis.org

1, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000
Offset: 1

Views

Author

Jaroslav Krizek, May 16 2022

Keywords

Comments

Sequence of different values of A000203(A245015(n)).
Conjecture: number 2 is the only factorial that is not in this sequence.

Examples

			Number 24 is in the sequence because sigma(14) = sigma(15) = sigma(23) = 24.

Crossrefs

Intersection of A000142 and A002191.
Cf. A000203, A338112, A055486, A055488, A245015, A354072, A354073, A338112.

Extensions

More terms from Jinyuan Wang, May 17 2022
Showing 1-4 of 4 results.

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