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%I #20 Oct 23 2017 23:15:53
%S 1,1,1,2,1,1,1,6,2,1,1,4,1,1,1,24,1,4,1,4,1,1,1,36,2,1,6,4,1,1,1,120,
%T 1,1,1,96,1,1,1,36,1,1,1,4,4,1,1,576,2,4,1,4,1,36,1,36,1,1,1,16,1,1,4,
%U 720,1,1,1,4,1,1,1,8640,1,1,4,4,1,1,1,576,24,1,1,16,1,1,1,36,1,16,1,4,1,1,1,14400,1,4,4,96,1,1,1,36,1
%N If n = p_1^e_1 * ... * p_k^e_k, p_1, ..., p_k primes, then a(n) = Product c! where c ranges over products of all combinations of exponents e_1, ..., e_k as {e_1, e_1*e_2, e_1*e_3, e_2*e_3, e_1*e_2*e_3, ..., e_1*e_2*...*e_k}.
%C a(1) = 1 (an empty product).
%H Antti Karttunen, <a href="/A293902/b293902.txt">Table of n, a(n) for n = 1..16384</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F For n = p^k * q * ... * r (with only one of the prime factors occurring multiple times), a(n) = A000142(k)^(2^(A001221(n)-1)).
%F a(p^n) = A000142(n), for any prime p.
%F For n > 1, a(n) = A163820(n) / A293900(n).
%e For n = 36 = 2^2 * 3^2 the combinations of the exponents are [], [2] (as exponent of 2), [2] (as exponent of 3) and [2, 2]. Taking products of these multisets we get 1 (as an empty product), 2, 2 and 4. Thus a(36) = 1! * 2! * 2! * 4! = 1*2*2*24 = 96.
%e For n = 72 = 2^3 * 3^2 the combinations of the exponents are [], [2], [3] and [2, 3]. Taking products of these multisets we get 1, 2, 3 and 6. Thus a(72) = 1! * 2! * 3! * 6! = 1*2*6*720 = 8640.
%t Array[Apply[Times, Map[Times @@ # &, Subsets@ FactorInteger[#][[All, -1]]]!] &, 105] (* _Michael De Vlieger_, Oct 23 2017 *)
%o (PARI)
%o A293902(n) = { my(exp_combos=powerset(factor(n)[, 2]), m=1); for(i=1,#exp_combos,m *= vecproduct(exp_combos[i])!); m; };
%o vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
%o powerset(v) = { my(siz=2^length(v),pv=vector(siz)); for(i=0,siz-1,pv[i+1] = choosebybits(v,i)); pv; };
%o choosebybits(v,m) = { my(s=vector(hammingweight(m)),i=j=1); while(m>0,if(m%2,s[j] = v[i];j++); i++; m >>= 1); s; };
%o (Scheme) (define (A293902 n) (if (= 1 n) n (/ (A163820 n) (A293900 n))))
%Y Cf. A000142, A163820, A293900.
%K nonn
%O 1,4
%A _Antti Karttunen_, Oct 22 2017
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