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A138534 Super least prime signatures; LCM of all signatures with n factors.

%I A138534 #27 Jul 02 2023 02:18:29
%S A138534 1,2,12,120,5040,110880,43243200,1470268800,1173274502400,
%T A138534 269853135552000,516498901446528000,32022931889684736000,
%U A138534 3234636350177055183360000,265240180714518525035520000,1163343432613878250805790720000,6014485546613750556665938022400000
%N A138534 Super least prime signatures; LCM of all signatures with n factors.
%C A138534 Also the row product of the following table:
%C A138534      1
%C A138534      2
%C A138534      4   3
%C A138534      8   3   5
%C A138534     16   9   5   7
%C A138534     32   9   5   7   11
%C A138534     64  27  25   7   11   13
%C A138534    128  27  25   7   11   13   17
%C A138534    256  81  25  49   11   13   17   19
%C A138534    512  81 125  49   11   13   17   19   23
%C A138534   1024 243 125  49  121   13   17   19   23   29
%C A138534   ...
%H A138534 Alois P. Heinz, <a href="/A138534/b138534.txt">Table of n, a(n) for n = 0..140</a>
%H A138534 Angelo B. Mingarelli, <a href="http://nntdm.net/volume-19-2013/number-4/43-76/">Abstract factorials</a>, Notes on Number Theory and Discrete Mathematics, Vol. 19, No. 4 (2013), pp. 43-76; <a href="https://arxiv.org/abs/0705.4299">arXiv preprint</a>, arXiv:0705.4299 [math.NT], 2007-2012.
%F A138534 From _Amiram Eldar_, Jul 02 2021: (Start)
%F A138534 a(n) = Product_{k=1..n} prime(k)^floor(n/k).
%F A138534 A001222(a(n)) = A006218(n). (End)
%F A138534 Sum_{n>=0} 1/a(n) = A346044. - _Amiram Eldar_, Jul 02 2023
%e A138534 For n = 3 the signatures are {8, 12, 30} so a(3) = 120.
%p A138534 b:= proc(n, i) option remember; `if`(n=0 or i<2, 2^n,
%p A138534       ilcm(seq(b(n-i*j, i-1)*ithprime(i)^j, j=0..n/i)))
%p A138534     end:
%p A138534 a:= n-> b(n$2):
%p A138534 seq(a(n), n=0..17);  # _Alois P. Heinz_, May 15 2015
%t A138534 b[n_, i_] := b[n, i] = If[n == 0 || i < 2, 2^n, LCM @@ Table[b[n - i j, i - 1] Prime[i]^j, {j, 0, n/i}]];
%t A138534 a[n_] := b[n, n];
%t A138534 a /@ Range[0, 17] (* _Jean-François Alcover_, Nov 02 2020, after _Alois P. Heinz_ *)
%t A138534 a[n_] := Product[Prime[k]^Floor[n/k], {k, 1, n}]; Array[a, 16, 0] (* _Amiram Eldar_, Jul 02 2021 *)
%o A138534 (PARI) a(n) = prod(k=1, n, prime(k)^(n\k)); \\ _Michel Marcus_, Jul 03 2021
%Y A138534 Cf. A006218, A010786, A010786, A010766.
%Y A138534 Subsequence of A025487.
%Y A138534 LCM of terms in rows of A215366.
%Y A138534 Cf. A001222, A006218, A346044.
%K A138534 nonn
%O A138534 0,2
%A A138534 _Alford Arnold_, Mar 28 2008
%E A138534 More terms from _Reikku Kulon_, Oct 02 2008