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 A336616 Maximum divisor of n! with distinct prime multiplicities. 6
 1, 1, 2, 3, 24, 40, 720, 1008, 8064, 72576, 3628800, 5702400, 68428800, 80870400, 317011968, 118879488000, 1902071808000, 2487324672000, 44771844096000, 50039119872000, 1000782397440000, 21016430346240000, 5085976143790080000, 6156707963535360000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct. LINKS David A. Corneth, Table of n, a(n) for n = 0..589 FORMULA a(n) = A327498(n!). EXAMPLE The sequence of terms together with their prime signatures begins: 1: () 1: () 2: (1) 3: (1) 24: (3,1) 40: (3,1) 720: (4,2,1) 1008: (4,2,1) 8064: (7,2,1) 72576: (7,4,1) 3628800: (8,4,2,1) 5702400: (8,4,2,1) 68428800: (10,5,2,1) 80870400: (10,5,2,1) 317011968: (11,5,2,1) 118879488000: (11,6,3,2,1) MATHEMATICA Table[Max@@Select[Divisors[n!], UnsameQ@@Last/@If[#==1, {}, FactorInteger[#]]&], {n, 0, 15}] PROG (PARI) a(n) = { if(n < 2, return(1)); my(pr = primes(primepi(n)), res = pr[#pr]); for(i = 1, #pr, pr[i] = [pr[i], val(n, pr[i])] ); forstep(i = #pr, 2, -1, if(pr[i][2] < pr[i-1][2], res*=pr[i-1][1]^pr[i-1][2] ) ); res } val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Aug 25 2020 CROSSREFS A327498 is the version not restricted to factorials, with quotient A327499. A336414 counts these divisors. A336617 is the quotient n!/a(n). A336618 is the version for equal prime multiplicities. A130091 lists numbers with distinct prime multiplicities. A181796 counts divisors with distinct prime multiplicities. A327526 gives the maximum divisor of n with equal prime multiplicities. A336415 counts divisors of n! with equal prime multiplicities. Cf. A000005, A000110, A098859, A118914, A124010, A336424, A336500, A336568. Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416. Sequence in context: A032811 A092049 A257789 * A354278 A061778 A160667 Adjacent sequences: A336613 A336614 A336615 * A336617 A336618 A336619 KEYWORD nonn AUTHOR Gus Wiseman, Jul 29 2020 STATUS approved

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Last modified January 27 21:49 EST 2023. Contains 359849 sequences. (Running on oeis4.)