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A113309 a(n) = the number of finite sequences of positive integers {b(k)} where (product b(k)) * (sum b(k)) = n. Different orderings of the same sequence {b(k)} are not counted separately.

%I A113309 #19 Nov 03 2017 13:09:51
%S A113309 1,1,1,2,1,2,1,2,2,2,1,3,1,2,2,4,1,3,1,4,2,2,1,5,2,2,2,4,1,5,1,4,2,2,
%T A113309 2,7,1,2,2,5,1,5,1,4,3,2,1,9,2,3,2,4,1,6,2,7,2,2,1,8,1,2,4,7,2,5,1,4,
%U A113309 2,5,1,11,1,2,3,4,2,5,1,9,4,2,1,10,2,2,2,7,1,9,2,4,2,2,2,13,1,3,4,7,1,5,1,7
%N A113309 a(n) = the number of finite sequences of positive integers {b(k)} where (product b(k)) * (sum b(k)) = n. Different orderings of the same sequence {b(k)} are not counted separately.
%C A113309 Sequence's terms calculated by "Max".
%C A113309 First occurrence: 1, 4, 12, 16, 24, 54, 36, 60, 48, 84, 72, 108, 96, ..., . - _Robert G. Wilson v_, May 03 2006
%H A113309 Alois P. Heinz, <a href="/A113309/b113309.txt">Table of n, a(n) for n = 1..20000</a> (first 4096 terms from Antti Karttunen)
%F A113309 a(n) = 1 iff n = 1 or n is a prime. a(n) = 2 if n is a semiprime. - _Robert G. Wilson v_, May 03 2006
%F A113309 a(n) = Sum_{d|n} {number of partitions of d where product of parts = n/d}. - _Antti Karttunen_, Nov 03 2017
%e A113309 6 = (1*1*1*1*1*1) * (1+1+1+1+1+1) = (1*2) * (1+2). So a(6) = 2.
%t A113309 t = Table[1, {104}]; Do[k = 1; lmt = PartitionsP[n]; p = IntegerPartitions[n]; While[k < lmt, a = Plus @@ p[[k]]*Times @@ p[[k]]; If[a < 105, t[[a]]++ ]; k++ ], {n, 52}]; t (* _Robert G. Wilson v_, May 03 2006 *)
%o A113309 (Scheme)
%o A113309 (define (A113309 n) (let ((z (list 0))) (let loop ((k n)) (cond ((zero? k) (car z)) ((not (zero? (modulo n k))) (loop (- k 1))) (else (begin (fold_over_partitions_with_uplim_cut k 1 * (lambda (partprod) (if (= n (* k partprod)) (set-car! z (+ 1 (car z))))) (/ n k)) (loop (- k 1))))))))
%o A113309 (define (fold_over_partitions_with_uplim_cut m initval addpartfun colfun uplim) (let recurse ((m m) (b m) (n 0) (partition initval)) (cond ((zero? m) (colfun partition)) ((> partition uplim) #f) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (addpartfun i partition)) (if (< i (min b m)) (loop (+ 1 i)))))))) ;; This function is a modification of fold_over_partitions_of given in A000793.
%o A113309 ;; _Antti Karttunen_, Nov 03 2017
%Y A113309 Cf. A057567, A113308.
%K A113309 nonn
%O A113309 1,4
%A A113309 _Leroy Quet_, Oct 25 2005
%E A113309 More terms from _Robert G. Wilson v_, May 03 2006