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 A228311 Numbers n such that the sum of digits of n! 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 n! is approx. equal to sf(n) = (9/2)*(nd-nz), where nd=A034886(n) is the number of digits of n!, which is about (log(n/E)*n + log(2*n*Pi)/2)/log(10), and nz=A027868(n) is the number of trailing zeros of n!, which is sum(k = 1, 2, ..) floor(n/5^k). - Giovanni Resta, Aug 28 2013 a(9) > 2.235*10^9 - Hans Havermann, May 16 2014 21966 = 666 in base 60. - Jon Perry, Aug 07 2014 Also the sum of digits of 21966 is a factorial (along with 1 and 2). - Jon Perry, Aug 12 2014 n! has ~ n log_10(n) digits, so its digit sum is typically close to C*n*log_10(n) for some constant C. A random number around k has probability something like log(k)/(k log log(k)) of being a factorial, so the probability that the digit sum of n! is a factorial should be something like const/(n log log n). The sum of this diverges, so we should expect infinitely many terms in the sequence. - Robert Israel, Aug 08 2014 LINKS "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 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 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 October 17 06:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)