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A243499 Product of parts of integer partitions as enumerated in the table A125106.

%I A243499 #36 Apr 21 2024 22:23:32
%S A243499 1,1,2,1,3,2,4,1,4,3,6,2,9,4,8,1,5,4,8,3,12,6,12,2,16,9,18,4,27,8,16,
%T A243499 1,6,5,10,4,15,8,16,3,20,12,24,6,36,12,24,2,25,16,32,9,48,18,36,4,64,
%U A243499 27,54,8,81,16,32,1,7,6,12,5,18,10,20,4,24,15,30,8,45,16,32,3
%N A243499 Product of parts of integer partitions as enumerated in the table A125106.
%C A243499 This sequence and A341392 have the same set of values on intervals from 2^m to 2^(m+1) - 1 for m >= 0. - _Mikhail Kurkov_, Jun 18 2021 [verification needed]
%H A243499 Antti Karttunen, <a href="/A243499/b243499.txt">Table of n, a(n) for n = 0..8191</a>
%F A243499 Can also be obtained by mapping with an appropriate permutation from the products of parts of each partition computed for other enumerations similar to A125106:
%F A243499 a(n) = A227184(A006068(n)).
%F A243499 a(n) = A003963(A005940(n+1)).
%F A243499 a(n) = A243504(A163511(n)).
%F A243499 From _Mikhail Kurkov_, Jul 11 2021: (Start)
%F A243499 a(n) = (1 + A023416(n))*a(A053645(n)) for n > 0 with a(0) = 1.
%F A243499 a(2n+1) = a(n) for n >= 0.
%F A243499 a(2n) = A341392(2*A059894(n)) = a(n - 2^f(n)) + a(2n - 2^f(n)) = (2 + f(n))*a(n - 2^f(n)) for n > 0 with a(0)=1 where f(n) = A007814(n).
%F A243499 Sum_{k=0..2^n - 1} a(k) = A000110(n+1) for n >= 0.
%F A243499 a((4^n - 1)/3) = n! for n >= 0.
%F A243499 a(2^m*(2^n - 1)) = (m+1)^n for n >= 0, m >= 0. (End) [verification needed]
%o A243499 (Scheme)
%o A243499 (define (A243499 n) (let loop ((n n) (i 1) (p 1)) (cond ((zero? n) p) ((even? n) (loop (/ n 2) (+ i 1) p)) (else (loop (/ (- n 1) 2) i (* p i))))))
%Y A243499 Cf. A125106, A161511 (gives the corresponding sums), A227184, A003963, A243504, A006068, A005940, A163511, A000110, A007814, A023416, A053645, A329369 (similar recurrence), A341392.
%K A243499 nonn,look
%O A243499 0,3
%A A243499 _Antti Karttunen_, Jun 28 2014