A020549
a(n) = (n!)^2 + 1.
Original entry on oeis.org
2, 2, 5, 37, 577, 14401, 518401, 25401601, 1625702401, 131681894401, 13168189440001, 1593350922240001, 229442532802560001, 38775788043632640001, 7600054456551997440001, 1710012252724199424000001, 437763136697395052544000001
Offset: 0
- T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 147.
- F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 237-240.
- H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
- M. Le, On the Interesting Smarandache Product Sequences, Smarandache Notions Journal, Vol. 9, No. 1-2, 1998, 133-134.
- M. Le, The Primes in Smarandache Power Product Sequences, Smarandache Notions Journal, Vol. 9, No. 1-2, 1998, 96-97.
- G. C. Greubel, Table of n, a(n) for n = 0..250
- M. Fleuren, Smarandache Square Products.
- F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
- Apoloniusz Tyszka, On sets X, subset of N, whose finiteness implies that we know an algorithm which for every n, element of N, decides the inequality max (X) < n, (2019).
- Apoloniusz Tyszka, On ZFC-formulae phi(x) for which we know a non-negative integer n such that max({x, element of N, phi(x)}) <= n if the set {x, element of N, phi(x)} is finite, 2019.
- Eric Weisstein's World of Mathematics, Factorial
- Eric Weisstein's World of Mathematics, Smarandache Sequences
For smallest prime factor see
A282706.
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with(combinat):seq(fibonacci(3,n!), n=0..16); # Zerinvary Lajos, Apr 21 2008
[seq(n!^2+1,n=0..20)]; # N. J. A. Sloane, Feb 26 2017
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Table[(n!)^2 + 1, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)
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a(n)=n!^2 + 1 \\ Charles R Greathouse IV, Nov 30 2016
A083341
Smaller factor of the n-th semiprime of the form (m!)^2 + 1.
Original entry on oeis.org
13, 101, 17, 101, 1344169, 149, 9049, 37, 710341, 2122590346576634509, 171707860473207588349837, 7686544942807799800864250520468090636146175134909, 2196283505473, 598350346949, 1211221552894876996541369232623365900407018851538797
Offset: 1
a(1) = 13 because (A083340(1)!)^2 + 1 = 518401 = 13*39877.
a(15) = 1211221552894876996541369232623365900407018851538797 because (A083340(15)!)^2 + 1 = (55!)^2 + 1 can be factored into P52*P96 with a(15) = P52.
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for(n=1,29,my(f=(n!)^2+1);if(bigomega(f)==2,print1(vecmin(factor(f)[,1]),", "))) \\ Hugo Pfoertner, Jul 13 2019
The 11th term of the sequence (49-digit factor of the 100-digit number (41!)^2+1) was found with Yuji Kida's multiple polynomial quadratic sieve UBASIC PPMPQS v3.5 in 13 days CPU time on an Intel PIII 550 MHz.
Missing a(4) and new a(14), a(15) added by
Hugo Pfoertner, Jul 13 2019
Original entry on oeis.org
3, 11, 59, 419, 31, 19, 1021019, 19399379, 41, 229, 173, 14841476269619, 919, 21519583, 53, 461, 151, 101, 281, 1115881660253397921934830779, 158616683341, 6435289534681345815798169108259, 271, 300423271
Offset: 1
- T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 147.
A083340
Numbers n such that A020549(n)=(n!)^2+1 is a semiprime.
Original entry on oeis.org
6, 7, 8, 12, 15, 16, 17, 18, 19, 28, 29, 41, 45, 53, 55, 61, 73
Offset: 1
a(1)=6 because (6!)^2+1=518401=13*39877 is a semiprime.
A301346
Largest prime factor of A020549(n) = (n!)^2 + 1.
Original entry on oeis.org
2, 2, 5, 37, 577, 14401, 39877, 251501, 95629553, 131681894401, 13168189440001, 1593350922240001, 2271708245569901, 38775788043632640001, 2404319663572286441, 1272170577304043929, 2938007628841577533852349, 13980942259426143240713449, 1107848353183710355135404972973, 20831587158104092560535861261
Offset: 0
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a:= n-> max(numtheory[factorset](n!^2+1)):
seq(a(n), n=0..20); # Alois P. Heinz, Mar 19 2018
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a(n) = vecmax(factor(n!^2 + 1)[,1]); \\ Daniel Suteu, Jun 10 2022
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