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A113309
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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.
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4
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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6 = (1*1*1*1*1*1) * (1+1+1+1+1+1) = (1*2) * (1+2). So a(6) = 2.
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MATHEMATICA
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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 *)
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PROG
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(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.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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