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A228311 Numbers k such that the sum of digits of k! is itself a factorial. 1
0, 1, 2, 3, 4, 21966, 176755, 182624820 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The sum of digits of k! is approximately (9/2)*(d-z), where d=A034886(k) is the number of digits of k!, which is about (log(k/E)*k + log(2*k*Pi)/2)/log(10), and z=A027868(k) is the number of trailing zeros of k!, which is Sum_{j=1, 2, ...) floor(k/5^j). - Giovanni Resta, Aug 28 2013

a(9) > 2.235*10^9 - Hans Havermann, May 16 2014

k! has ~ k log_10(k) digits, so its digit sum is typically close to C*k*log_10(k) for some constant C. A random number around j has probability something like log(j)/(j log log(j)) of being a factorial, so the probability that the digit sum of k! is a factorial should be something like const/(k log log k). The sum of this diverges, so we should expect infinitely many terms in the sequence. - Robert Israel, Aug 08 2014

LINKS

Table of n, a(n) for n=1..8.

"Mouhaha" Digit sums and factorials

EXAMPLE

The sum of the digits of 21966! is 362880 = 9!.

The sum of the digits of 176755! is 3628800 = 10!.

The sum of the digits of 182624820! is 6227020800 = 13!.

MATHEMATICA

lst = {0}; k = p = 1; fctl = Range@ 15!; While[k < 180000, p = p*k; While[ Mod[p, 10] == 0, p /= 10]; If[ MemberQ[ fctl, Plus @@ IntegerDigits@ p], Print[k]; AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Feb 18 2014 *)

PROG

(PARI) lpf(n)=my(f=factor(n)[, 1]); f[1]

factorial_lval(n, p)={

    my(s);

    while(n\=p, s+=n);

    s

};

isfactorial(n)={

    if(n<3, return(n>0));

    my(v2=valuation(n, 2), mn=v2+1, mx=mn+log(v2+1.5)\log(2), t, c);

    while (mx - mn > 1,

        t = mn + (mx - mn)\2;

        c = factorial_lval(t, 2);

        if (c < v2,

            mn = t+1

        ,

            if (c > v2,

                mx = t-1

            ,

                mx = bitor(t, 1);

                mn = max(mn, mx-1)

            )

        )

    );

    if (mn < mx,

        my(p=lpf(mx));

        t = valuation(n, p);

        c = factorial_lval(mx, p);

        if (t > c, return(0));

        if (t == c,

            mn = mx

        )

    );

    n==mn!

};

is(n)=isfactorial(sumdigits(n!))

CROSSREFS

Cf. A229024, A004152.

Sequence in context: A062929 A038105 A143716 * A258107 A307256 A107656

Adjacent sequences:  A228308 A228309 A228310 * A228312 A228313 A228314

KEYWORD

nonn,base,hard,nice,more

AUTHOR

Charles R Greathouse IV, Aug 27 2013

EXTENSIONS

a(8) from Hans Havermann, Mar 24 2014

STATUS

approved

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Last modified August 14 17:12 EDT 2022. Contains 356122 sequences. (Running on oeis4.)