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A336415 Number of divisors of n! with equal prime multiplicities.

%I A336415 #42 Nov 18 2021 12:45:49
%S A336415 1,1,2,4,6,10,13,21,24,28,33,49,53,85,94,100,104,168,173,301,307,317,
%T A336415 334,590,595,603,636,642,652,1164,1171,2195,2200,2218,2283,2295,2301,
%U A336415 4349,4478,4512,4519,8615,8626,16818,16836,16844,17101,33485,33491,33507,33516,33582
%N A336415 Number of divisors of n! with equal prime multiplicities.
%C A336415 A number k has "equal prime multiplicities" (or is "uniform") iff its prime signature is constant, meaning that k is a power of a squarefree number.
%H A336415 David A. Corneth, <a href="/A336415/b336415.txt">Table of n, a(n) for n = 0..9999</a>
%H A336415 Jon Maiga, <a href="http://sequencedb.net/s/A336415">Computer-generated formulas for A336415</a>, Sequence Machine.
%F A336415 a(n) = A327527(n!).
%e A336415 The a(n) uniform divisors of n for n = 1, 2, 6, 8, 30, 36 are the columns:
%e A336415   1  2  6  8  30  36
%e A336415      1  3  6  15  30
%e A336415         2  4  10  16
%e A336415         1  3   8  15
%e A336415            2   6  10
%e A336415            1   5   9
%e A336415                4   8
%e A336415                3   6
%e A336415                2   5
%e A336415                1   4
%e A336415                    3
%e A336415                    2
%e A336415                    1
%e A336415 In 20!, the multiplicity of the third prime (5) is 4 but the multiplicity of the fourth prime (7) is 2. Hence there are 2^3 - 1 = 3 divisors with all exponents 3 (we subtract |{1}| = 1 from that count as 1 has no exponent 3). - _David A. Corneth_, Jul 27 2020
%t A336415 Table[Length[Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&]],{n,0,15}]
%o A336415 (PARI) a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); (#ex==0) || (vecmin(ex) == vecmax(ex))); \\ _Michel Marcus_, Jul 24 2020
%o A336415 (PARI) a(n) = {if(n<2, return(1)); my(f = primes(primepi(n)), res = 1, t = #f); f = vector(#f, i, val(n, f[i])); for(i = 1, f[1], while(f[t] < i, t--; ); res+=(1<<t - 1) ); res }
%o A336415 val(n, p) = my(r=0); while(n, r+=n\=p);r \\ _David A. Corneth_, Jul 27 2020
%Y A336415 The version for distinct prime multiplicities is A336414.
%Y A336415 The version for nonprime perfect powers is A336416.
%Y A336415 Uniform partitions are counted by A047966.
%Y A336415 Uniform numbers are A072774, with nonprime terms A182853.
%Y A336415 Numbers with distinct prime multiplicities are A130091.
%Y A336415 Divisors with distinct prime multiplicities are counted by A181796.
%Y A336415 Maximum divisor with distinct prime multiplicities is A327498.
%Y A336415 Uniform divisors are counted by A327527.
%Y A336415 Maximum uniform divisor is A336618.
%Y A336415 1st differences are given by A048675.
%Y A336415 Cf. A000005, A000961, A001222, A001597, A007916, A112798, A124010, A327499.
%Y A336415 Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617.
%K A336415 nonn
%O A336415 0,3
%A A336415 _Gus Wiseman_, Jul 22 2020
%E A336415 Terms a(31) and onwards from _David A. Corneth_, Jul 27 2020