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.
[a: n in [1..100] | IsPrime(a) where a is Factorial(n) div 10^Valuation(Factorial(n), 5)+1]; // Vincenzo Librandi, Jun 03 2016
A004154 := proc(n) option remember ; local m ; if n <= 1 then RETURN(1) ; else m := A004154(n-1)*n ; while m mod 10 = 0 do m := m /10 ; end ; RETURN(m) ; fi ; end: n := 1 ; while true do f := A004154(n) : if isprime(f-1) then printf("%d, ",f-1) ; fi ; n := n+1 : od : # R. J. Mathar, Feb 27 2007
v=[];for(n=1,500,if(ispseudoprime(t=n!/10^valuation(n!,5)-1),v=concat(v,t))); v \\ Charles R Greathouse IV, Feb 14 2011
f[n_] := n!/10^Sum[ Floor[n/5^k], {k, 1, Log[10, n] + 1}]; Do[ If[ PrimeQ[ f[n] + 1], Print[n]], {n, 1, 850}]
f[n_] := n!/10^Sum[ Floor[n/5^k], {k, 1, Log[10, n] + 1}]; Do[ If[ PrimeQ[ f[n] - 1], Print[n]], {n, 1, 750}]
a[n_] := Mod[n!/10^IntegerExponent[n!, 10], NextPrime[n]]; Array[a, 69, 0](* Becomes quicker as n increases and it uses less resources. For me, this is around 2 million *)g[n_, p_] := Block[{s = 0, e = 1}, While[t = Floor[n/p^e]; t > 0, s += t; e++]; s];f[n_] := Block[{m = NextPrime@ n, p = 1, q = 7}, p = PowerMod[2, g[n, 2] - g[n, 5], m]; p = Mod[p*PowerMod[3, g[n, 3], m], m]; While[q < n +1, p = Mod[p*PowerMod[q, g[n, q], m], m]; q = NextPrime@ q]; p]
from functools import reduce from sympy import nextprime from sympy.ntheory.factor_ import digits def A376286(n): return ((p:=nextprime(n))-1)*pow(reduce(lambda i, j:i*j%p, range(n+1,p),1),-1,p)*pow(10,sum(digits(n,5)[1:])-n>>2,p)%p # Chai Wah Wu, Oct 18 2024
f[n_] := n!/10^Sum[Floor[n/5^k], {k, 1, Log[10, n] + 1}]; Do[ If[ PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ f[n]]]] + 1], Print[n]], {n, 0, 800}]
f[n_] := n!/10^Sum[Floor[n/5^k], {k, 1, Log[10, n] + 1}]; Do[ If[ PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ f[n]]]] - 1], Print[n]], {n, 1, 800}]
a(100) = 24. a(10^3) = 249. a(10^4) = 2499. a(10^5) = 24999. a(10^6) = 249998. a(10^7) = 2499999. a(10^8) = 24999999. a(10^9) = 249999998. a(10^n) = 10^n/4 - 3 for 10 <= n <= 15 except for a(10^14) = 10^14/4 - 2. - _M. F. Hasler_, Dec 27 2019
a027868 n = sum $ takeWhile (> 0) $ map (n `div`) $ tail a000351_list -- Reinhard Zumkeller, Oct 31 2012
[Valuation(Factorial(n), 5): n in [0..80]]; // Bruno Berselli, Oct 11 2021
0, seq(add(floor(n/5^i),i=1..floor(log[5](n))), n=1..100); # Robert Israel, Nov 13 2014
Table[t = 0; p = 5; While[s = Floor[n/p]; t = t + s; s > 0, p *= 5]; t, {n, 0, 100} ]
Table[ IntegerExponent[n!], {n, 0, 80}] (* Robert G. Wilson v *)
zOF[n_Integer?Positive]:=Module[{maxpow=0},While[5^maxpow<=n,maxpow++];Plus@@Table[Quotient[n,5^i],{i,maxpow-1}]]; Attributes[zOF]={Listable}; Join[{0},zOF[ Range[100]]] (* Harvey P. Dale, Apr 11 2022 *)
a(n)={my(s);while(n\=5,s+=n);s} \\ Charles R Greathouse IV, Nov 08 2012, edited by M. F. Hasler, Dec 27 2019
a(n)=valuation(n!,5) \\ Charles R Greathouse IV, Nov 08 2012
apply( A027868(n)=(n-sumdigits(n,5))\4, [0..99]) \\ M. F. Hasler, Dec 27 2019
from sympy import multiplicity A027868, p5 = [0,0,0,0,0], 0 for n in range(5,10**3,5): p5 += multiplicity(5,n) A027868.extend([p5]*5) # Chai Wah Wu, Sep 05 2014
def A027868(n): return 0 if n<5 else n//5 + A027868(n//5) # David Radcliffe, Jun 26 2016
from sympy.ntheory.factor_ import digits def A027868(n): return n-sum(digits(n,5)[1:])>>2 # Chai Wah Wu, Oct 18 2024
6! = 720, so a(6) = 2.
a008904 n = a008904_list !! n
a008904_list = 1 : 1 : f 2 1 where
f n x = x' `mod` 10 : f (n+1) x' where
x' = g (n * x) where
g m | m `mod` 5 > 0 = m
| otherwise = g (m `div` 10)
-- Reinhard Zumkeller, Apr 08 2011
f[n_]:=Module[{m=n!},While[Mod[m,10]==0,m=m/10];Mod[m,10]]
Table[f[i],{i,0,100}]
f[n_] := Mod[6Times @@ (Rest[FoldList[{ 1 + #1[[1]], #2!2^(#1[[1]]#2)} &, {0, 0}, Reverse[IntegerDigits[n, 5]]]]), 10][[2]]; Join[{1, 1}, Table[f[n], {n, 2, 100}]] (* program contributed by Jacob A. Siehler, Greg Dresden, Feb 21 2006 *)
zOF[n_Integer?Positive] := Module[{maxpow=0}, While[5^maxpow<=n,maxpow++]; Plus@@Table[Quotient[n,5^i], {i,maxpow-1}]]; Flatten[Table[ Take[ IntegerDigits[ n!], {-zOF[n]-1}],{n,100}]] (* Harvey P. Dale, Dec 16 2010 *)
f[n_]:=Block[{id=IntegerDigits[n!, 10]}, While[id[[-1]]==0, id=Most@id]; id[[-1]]]; Table[f@n, {n, 0, 100}] (* Vincenzo Librandi, Sep 07 2017 *)
a(n) = r=1; while(n>0, r *= Mod(4, 10)^((n\10)%2) * [1, 2, 6, 4, 2, 2, 4, 2, 8][max(n%10, 1)]; n\=5); lift(r) \\ Charles R Greathouse IV, Nov 05 2010; cleaned up by Max Alekseyev, Jan 28 2012
def a(n):
if n <= 1: return 1
return 6*[1,1,2,6,4,4,4,8,4,6][n%10]*3**(n/5%4)*a(n/5)%10
# Maciej Ireneusz Wilczynski, Aug 23 2010
from functools import reduce from sympy.ntheory.factor_ import digits def A008904(n): return reduce(lambda x,y:x*y%10,(((6,2,4,8,6,2,4,8,2,4,8,6,6,2,4,8,4,8,6,2)[(a<<2)|(i*a&3)] if i*a else (1,1,2,6,4)[a]) for i, a in enumerate(digits(n,5)[-1:0:-1])),6) if n>1 else 1 # Chai Wah Wu, Dec 07 2023
def A008904(n): # algorithm from David Wilson, http://oeis.org/A008904/a008904b.txt if n == 0 or n == 1: return 1 dd = n.digits(base=5) x = sum(i*d for i,d in enumerate(dd)) y = sum(d for d in dd if d % 2 == 0)/2 z = 2**((x+y) % 4) if z == 1: z = 6 return z # D. S. McNeil, Dec 09 2010
From _David A. Corneth_, Jun 15 2024: (Start) a(1) = 2. a(2) = 0 or 1. If a(2) = 0 then any number ending in 02 (the backwards concatenation of (2, 0)) should be a multiple of 2^2 = 4 but it is not. Any number ending in 12 (the backwards concatenation of (2, 1)) is a multiple of 2^2 = 4 so a(2) = 1. Similarly a(3) = 1 as 112 is a multiple of 2^3 and 012 is not. a(4) = 0 as 0112 is a multiple of 2^4 and 1112 is not. a(5) = 1 as 10112 is a multiple of 2^5 and 00112 is not. (End)
first(n) = {
my(t = 2);
for(i = 2, n,
if(t%2^i != 0,
t = t + 10^(i-1);
);
);
res = Vecrev(digits(t));
res = concat(res, vector(n - #res));
} \\ David A. Corneth, Jun 15 2024
a,b=2,1; print('2',end=',')
for i in range(200): a=(a+(a&1)*b)>>1; b+=b<<2; print(a&1,end=',')
# Cezary Glowacz, Apr 12 2025
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