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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 *) With[{fcts=Range[20]!}, Select[Range[0, 22000], MemberQ[fcts, Total[IntegerDigits[#!]]]&]] (* Harvey P. Dale, Jan 06 2024 *) 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 February 21 02:25 EST 2024. Contains 370219 sequences. (Running on oeis4.)