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

1,4

Comments

Sequence's terms calculated by "Max".

First occurrence: 1, 4, 12, 16, 24, 54, 36, 60, 48, 84, 72, 108, 96, ..., . - Robert G. Wilson v, May 03 2006

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 4096 terms from Antti Karttunen)

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

Example

6 = (1*1*1*1*1*1) * (1+1+1+1+1+1) = (1*2) * (1+2). So a(6) = 2.

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 *)

Prog

(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

Crossrefs

Cf. A057567, A113308.

Sequence in context: A068108 A342679 A337135 * A348955 A062362 A330437

Adjacent sequences: A113306 A113307 A113308 * A113310 A113311 A113312

Keyword

nonn

Author

Leroy Quet, Oct 25 2005

Extensions

More terms from Robert G. Wilson v, May 03 2006

Status

approved