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A055487
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Least m such that phi(m) = n!.
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8
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1, 3, 7, 35, 143, 779, 5183, 40723, 364087, 3632617, 39916801, 479045521, 6227180929, 87178882081, 1307676655073, 20922799053799, 355687465815361, 6402373865831809, 121645101106397521, 2432902011297772771
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Erdos believed (see Guy reference) that Phi[x] = n! is solvable.
Factorial primes of p = A002981[m]!+1 = k!+1 form give 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 (joseph_l_pe(AT)hotmail.com), Oct 01 2002
A123476(n) is a solution to the equation phi(x)=n! - T. D. Noe (noe(AT)sspectra.com), Sep 27 2006
Contribution from M. F. Hasler (www.univ-ag.fr/~mhasler), 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 <= ceil(sqrt(n!))". The case "= ceil(...)" occurs for n=5, sqrt(120)=10.95..., p=11, q=13.
A055487(n) is the first element in row n of the table A165773, which lists all solutions to phi(x)=n!. Thus A055487(n)=A165773(sum(A055506(k),k<n)+1). The last element of each row (i.e. the largest solution to phi(x)=n!) is given in A165774. (End)
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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.
P. Erdos and J. Lambek, Problem 4221, Amer. Math. Monthly, 55 (1948), 103.
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FORMULA
| a(n) = Min{m : phi(m) = n!} = Min{m : A000010(m) = A000142(n)}
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PROG
| (PARI) A055487(n)={ my( f=n!, p=sqrtint(f)); isprime(f+1) && return(f+(n>1)); until( isprime(f/p+1), while( f%p=nextprime(p+2)-1, )); (p+1)*(f/p+1) } /* based on the conjecture */ [From M. F. Hasler (www.univ-ag.fr/~mhasler), Oct 04 2009]
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CROSSREFS
| Cf. A055486-A055489, A055506, A000010, A000142.
Sequence in context: A047907 A145874 A147681 * A121130 A006099 A053530
Adjacent sequences: A055484 A055485 A055486 * A055488 A055489 A055490
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KEYWORD
| nonn
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AUTHOR
| Labos E. (labos(AT)ana.sote.hu), Jun 28 2000
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EXTENSIONS
| More terms from djr(AT)nk.ca, Nov 05 2001
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