This paper presents an effective method for calculating !n, and implements the
Smarandache-Kurepa function by using one new method

Perry, Jon

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AN IMPROVED ALGORITHM FOR CALCULATINGTHE SUM-OF-FACTORIALS FUNCTIONJon Perry51 High Street, Belbroughton, West Midlands, DY9 9ST, EnglandAbstract The sum of factorials function, also known as the left factorial function, is de-fined as !n = 0! + 1! + · · ·+ (n− 1)!. These have been used by Smarandacheand Kurepa to define the Smarandache-Kurepa Function (see reference [1], [2]).This paper presents an effective method for calculating !n, and implements theSmarandache-Kurepa function by using one new method.1. IntroductionWe define !n as 0! + 1! + · · ·+ (n− 1)!.A simple PARI/GP program to calculate these values is below:soff(n) =n−1∑i=0i!Then,for(i = 0, 10,print1(","soff(i))) gives the desired output;0, 1, 2, 4, 10, 34, 154, 874, 5914, 46234, 409114,which is A003422 at OEIS [3].2. A new methodIf we write out what the sum of factorials function is doing, we can write:1+1+1.2+1.2.3+1.2.3.4+1.2.3.4.5+and so on.If we now read down the columns, we see that this can be written as:1 + 1[1 + 2[1 + 3[1 + 4[1 + · · · .76 SCIENTIA MAGNA VOL.1, NO.1This is because we have an opening 1 from 0!. Then 1 is a factor of all theremaining factorials. However 1 is the only factor of 1 of the factorials, namely1!, so we have1 + 1[1 + · · · .Having removed the 1!, 2 is now a factor is all remaining factorials, and isthe final factor in 2!, hence1 + 1[1 + 2[1 + · · ·and so on.!n requires inputs from 0! to (n− 1)!, and hence we are required to stop thenested recursion by n− 1. e.g. for !5, we have1 + 1[1 + 2[1 + 3[1 + 4[1]]]].We can validate this:1 + 1[1 + 2[1 + 3[1 + 4[1]]]]= 1 + 1[1 + 2[1 + 3[5]]]= 1 + 1[1 + 2[16]]= 1 + 1[33]= 34.3. Code for new methodWe can see how the new method decreases execution time, the originalmethod presented performs O(k2) multiplications and O(k) additions. Thismethod performs O(k) multiplications and O(k) additions.PARI/GP code for the routine is below:qsoff(n) = local(r); r = n; forstep(i = n− 2, 1, 4− 1, r∗ = i; r ++); r4. Implementing the Smarandache-Kurepa functionWe need only consider primes, and the sk variable needs only range from 1to p− 1 (if !1 to !p are not divisible by p, then !(p+ k) will never be as all newterms have p as a factor).For prime (p = 2, 500, for (sk = 1, p, if (qsoff(sk)This is obviously wasteful, we are calculating qsoff(sk) very repetitively.The code below stores the qsoff() values in a vector.An Improved Algorithm for Calculating the Sum-of-Factorials Function 77v=vector(500, i, qsoff(i)); forprime (p = 2500, for (sk = 1, p, if (v[sk]The following output is produced:2,−, 4, 6, 6,−, 5, 7, 7,−, 12, 22, 16,−,−,−,−, 55,−, 54, 42,−,−, 24,−,−, 25,−,−, 86,−, 97,−, 133,−,−, 64, 94, 72, 58,−,−, 49, 69, 19,−, 78,−, 14,−, 208,167,−, 138, 80, 59,−,−,−,−, 63, 142, 41,−, 110, 22, 286, 39,−, 84,−,−, 215, 80,14, 305,−, 188, 151, 53, 187,−, 180,−,−,−,−, 44, 32, 83, 92,−, 300, 16,−.5. Additional relationsThe basic pattern created in this paper also allows for the rapid calculationof other Smarandache-like functions based on the sum of factorials function.For example, we could define SSF (n) as the sum of squares factorial, e.g.SSF (10) = 0! + 1! + 4! + 9!, and the corresponding general expansion is1 + 1[1 + 2.3.4[1 + 5.6.7.8.9[1 + · · · .Or we can define the sum of factorials squared function as0!2 + 1!2 + 2!2 + · · · .In this case, the expansion is1 + 1[1 + 4[1 + 9[1 + · · · .References[1] Smarandache-Kurepa Function: http://www.gallup.unm.edu/ smarandache/FUNCT1.TXT[2] Eric W. Weisstein. "Smarandache-Kurepa Function." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Smarandache-KurepaFunction.html[3] OEIS : A003422 (Left factorials) http://www.research.att.com/projects/OEISAnum=A003422

AN IMPROVED ALGORITHM FOR CALCULATING THE SUM-OF-FACTORIALS FUNCTION

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

https://zenodo.org/record/9164/files/ImprovedAlgorthm.pdf

AN IMPROVED ALGORITHM FOR CALCULATING THE SUM-OF-FACTORIALS FUNCTION

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