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.
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, 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, 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
Offset: 1
Keywords
Examples
6 = (1*1*1*1*1*1) * (1+1+1+1+1+1) = (1*2) * (1+2). So a(6) = 2.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 4096 terms from Antti Karttunen)
Programs
-
Mathematica
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 *) -
Scheme
(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)))))))) (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. ;; Antti Karttunen, Nov 03 2017
Formula
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
a(n) = Sum_{d|n} {number of partitions of d where product of parts = n/d}. - Antti Karttunen, Nov 03 2017
Extensions
More terms from Robert G. Wilson v, May 03 2006
Comments