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find a number such that its sum of factorial of its digits is equal to 100

find a number such that its sum of factorial of its digits is equal to 100

Grade:12

3 Answers

Har Simrat Singh
42 Points
8 years ago

there can be many such numbers like a number with 4times 4 and 2 times 2 like 444422

and 444333322 and all their possible rearrangements but one thing is for sure the digits will  be 1,2,3,4 only because 5fact is 120

azeem khan
32 Points
8 years ago

the no.is-  444224

yours katarnak Suresh
43 Points
8 years ago

Factorial Sums

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The sum-of-factorials function is defined by

S_1(n) = sum_(k=1)^(n)k!
(1)
= (-e+Ei(1)+pii+E_(n+2)(-1)Gamma(n+2))/e
(2)
= (-e+Ei(1)+R[E_(n+2)(-1)]Gamma(n+2))/e,
(3)

where Ei(z) is the exponential integral, Ei(1) approx 1.89512 (Sloane''s A091725), E_n is the En-function, R[z] is the real part of z, and i is the imaginary number. The first few values are 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, ... (Sloane''s A007489). S_1(n) cannot be written as a hypergeometric term plus a constant (Petkovšek et al. 1996).

The related sum with index running from 0 instead of 1 is sometimes denoted L!n (not to be confused with the subfactorial) and known as the left factorial,

 L!n=sum_(k=0)^nk!.
(4)

The related sum with alternating terms is known as the alternating factorial,

 a(n)=sum_(k=1)^n(-1)^(n-k)k!.
(5)

The sum

S_2(n) = sum_(k=1)^(n)kk!
(6)
= (n+1)!-1
(7)

has a simple form, with the first few values being 1, 5, 23, 119, 719, 5039, ... (Sloane''s A033312).

Identities satisfied by sums of factorials include

sum_(k=0)^(infty)1/(k!) = e=2.718281828...
(8)
sum_(k=0)^(infty)((-1)^k)/(k!) = e^(-1)=0.3678794411...
(9)
sum_(k=0)^(infty)1/((k!)^2) = I_0(2)=2.279585302...
(10)
sum_(k=0)^(infty)((-1)^k)/((k!)^2) = J_0(2)=0.2238907791...
(11)
sum_(k=0)^(infty)1/((2k)!) = cosh1=1.543080634...
(12)
sum_(k=0)^(infty)((-1)^k)/((2k)!) = cos1=0.5403023058...
(13)
sum_(k=0)^(infty)1/((2k+1)!) = sinh1=1.175201193...
(14)
sum_(k=0)^(infty)((-1)^k)/((2k+1)!) = sin1=0.8414709848...
(15)

(Sloane''s A001113, A068985, A070910, A091681, A073743, A049470, A073742, and A049469; Spanier and Oldham 1987), where I_0(x) is a modified Bessel function of the first kind, J_0(x) is a Bessel function of the first kind, coshx is the hyperbolic cosine, cosx is the cosine, sinhx is the hyperbolic sine, and sinx is the sine.

The sum

 S_3(n)=sum_(k=1)^n(k!)^2
(16)

does not appear to have a simple closed form, but its values for n=1, 2, ... are 1, 5, 41, 617, 15017, 533417, 25935017, ... (Sloane''s A104344). It is prime for indices 2, 3, 4, 5, 7, 8, 10, 18, 21, 42, 51, 91, 133, 177, 182, 310, 3175, 9566, ... (Sloane''s A100289). It is known that there can be only a finite number of such primes, but it is not known what the last term is.

Sums of factorial powers include

sum_(n=0)^(infty)((n!)^2)/((2n)!) = 2/(27)(18+sqrt(3)pi)
(17)
= 1.73639985...
(18)
sum_(n=0)^(infty)((n!)^3)/((3n)!) = _3F_2(1,1,1;1/3,2/3;1/(27))
(19)
= 1.17840325...
(20)

(Sloane''s A091682 and A091683) and, in general,

 sum_(n=0)^infty((n!)^k)/((kn)!)=_kF_(k-1)(1,...,1_()_(k);1/k,2/k,...,(k-1)/k;1/(k^k)).
(21)

Schroeppel and Gosper (1972) give the integral representation

 sum_(n=0)^infty((n!)^3)/((3n)!)=int_0^1[P(t)+Q(t)cos^(-1)R(t)]dt,
(22)

where

P(t) = (2(8+7t^2-7t^3))/((4-t^2+t^3)^2)
(23)
Q(t) = (4t(1-t)(5+t^2-t^3))/((4-t^2+t^3)^2sqrt((1-t)(4-t^2+t^3)))
(24)
R(t) = 1-1/2(t^2-t^3).
(25)

There are only four integers equal to the sum of the factorials of their digits. Such numbers are called factorions.

While no factorial greater than 1! is a square number, D. Hoey listed sums <10^(12) of distinct factorials which give square numbers, and J. McCranie gave the one additional sum less than 21!=5.1×10^(19):

0!+1!+2! = 2^2
(26)
1!+2!+3! = 3^2
(27)
1!+4! = 5^2
(28)
1!+5! = 11^2
(29)
4!+5! = 12^2
(30)
1!+2!+3!+6! = 27^2
(31)
1!+5!+6! = 29^2
(32)
1!+7! = 71^2
(33)
4!+5!+7! = 72^2
(34)
1!+2!+3!+7!+8! = 213^2
(35)
1!+4!+5!+6!+7!+8! = 215^2
(36)
1!+2!+3!+6!+9! = 603^2
(37)
1!+4!+8!+9! = 635^2
(38)
1!+2!+3!+6!+7!+8!+10! = 1917^2
(39)

and

 1!+2!+3!+7!+8!+9!+10!+11!+12!+13!+14!+15!=1183893^2
(40)

(Sloane''s A014597).

Sums with powers of an index in the numerator and products of factorials in the denominator can often be done analytically in terms of regularized hypergeometric functions _pF^~_q, for example

 sum_(k=0)^N1/((k+m)!(k+n)!)=_1F^~_2(1;m+1,n+1;1)   -_1F^~_2(1;m+N+2;n+N+2;1)   sum_(k=0)^N1/((m+k)!(n-k)!)=(_2F^~_1(1,-n;m+1;-1))/(Gamma(n+1))   -(_2F^~_1(1,-n+N+1;m+N+2;-1))/(Gamma(n-N)).
(41)

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