A139060
Primes of the form (4+k!)/4.
Original entry on oeis.org
7, 31, 181, 1556755201, 12772735542927360001, 3877802510832746496000001, 65782709233423382541804503040000001, 203978820811974433586402817399028973568000000001
Offset: 1
Cf.
A007749,
A020458,
A082672,
A089085,
A089130,
A117141,
A139056,
A139057,
A139058,
A139059,
A139061,
A139062,
A139063,
A139064,
A139065,
A139066,
A139151.
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a = {}; Do[If[PrimeQ[(n! + 4)/4], AppendTo[a, (n! + 4)/4]], {n, 1, 50}]; a
Select[(4+Range[100]!)/4,PrimeQ] (* Harvey P. Dale, Oct 05 2016 *)
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for(k=4,1e3,if(ispseudoprime(t=k!/4+1),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011
A139149
a(n) = (n!+2)/2.
Original entry on oeis.org
2, 4, 13, 61, 361, 2521, 20161, 181441, 1814401, 19958401, 239500801, 3113510401, 43589145601, 653837184001, 10461394944001, 177843714048001, 3201186852864001, 60822550204416001, 1216451004088320001, 25545471085854720001, 562000363888803840001
Offset: 2
(1!+2)/2 = 3/2 is not an integer.
a(2) = (2!+2)/2 = 2.
Offsets for above sequences are Kempner numbers
A002034.
For smallest number of the form (m!+n)/n see
A139148.
Cf.
A007749,
A020458,
A082672,
A089085,
A089130,
A117141,
A137390,
A139056-
A139066,
A139068,
A139070-
A139075,
A139157,
A139159-
A139162.
A181892
a(n) = (n!/4-1)^2.
Original entry on oeis.org
25, 841, 32041, 1585081, 101586241, 8229936961, 823010025601, 99584412681601, 14340158060659201, 2423486749613529601, 475003403490910694401, 106875765794608626816001, 27360196043576729389056001, 7907096656596520292861952001, 2561899316737326995063771136001
Offset: 4
A138196
Number of different ways n! can be represented as the difference of two squares; also, for n >= 4, half the number of positive integer divisors of n!/4.
Original entry on oeis.org
1, 0, 0, 2, 4, 9, 18, 36, 60, 105, 210, 324, 648, 1080, 1680, 2352, 4704, 6480, 12960, 18360, 27200, 43200, 86400, 110880, 155232, 243936, 310464, 423360, 846720, 1080000, 2160000, 2592000, 3686400, 5713920, 7713792, 9237888, 18475776
Offset: 1
John T. Robinson (jrobinson(AT)acm.org), May 04 2008
a(5)=4 since 5! = 120 = 31^2 - 29^2 = 17^2 - 13^2 = 13^2 - 7^2 = 11^2 - 1^2.
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A138196 := proc(n)
if n <= 3 then
op(n,[1,0,0]) ;
else
numtheory[tau](n!/4)/2 ;
end if;
end proc: # R. J. Mathar, Apr 03 2012
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(* for n>=4 *) cc = {}; Do[w = n!/4; kk = Floor[(DivisorSigma[0, w] + 1)/2]; AppendTo[cc, kk], {n, 4, 54}]; cc (* Artur Jasinski, Mar 31 2012 *)
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a(n) = if (n<4, (n==1), numdiv(n!/4)/2); \\ Michel Marcus, Jun 22 2019
A181893
Number of different nonnegative solutions of equation: x^2 - y^2 = k! for 1 <= k <= n.
Original entry on oeis.org
1, 1, 1, 3, 7, 16, 34, 70, 130, 235, 445, 769, 1417, 2497, 4177, 6529, 11233, 17713, 30673, 49033, 76233, 119433, 205833, 316713, 471945, 715881, 1026345, 1449705, 2296425, 3376425, 5536425, 8128425, 11814825, 17528745, 25242537, 34480425, 52956201, 81509673
Offset: 1
-
kkk = 1; cc = {1, 1, 1}; Do[w = n!/4; kk = Floor[(DivisorSigma[0, w] + 1)/2]; kkk = kkk + kk; AppendTo[cc, kkk], {n, 4, 54}]; cc
A181899
Largest divisor of n!/4 which is less than sqrt(n!)/2.
Original entry on oeis.org
2, 5, 12, 35, 96, 288, 945, 3150, 10800, 39312, 147420, 571536, 2286144, 9424800, 39984000, 174283200, 779688000, 3573570000, 16761064320, 80379048750, 393826406400, 1969132032000, 10040487256800, 52174220175000, 276080056560000, 1486750296281250
Offset: 4
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Table[f = n!/4; Select[Divisors[f], # <= Sqrt[f] &][[-1]], {n, 4, 20}]
Showing 1-6 of 6 results.
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