A293813 -id:A293813 - cp's OEIS frontend

A293814 Number of partitions of n into distinct nontrivial divisors of n.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 18, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 13, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 10, 0, 0, 0, 1

Offset: 0

Ilya Gutkovskiy, Oct 16 2017

nonn

			a(24) = 2 because 24 has 8 divisors {1, 2, 3, 4, 6, 8, 12, 24} among which 6 are nontrivial divisors {2, 3, 4, 6, 8, 12} therefore we have [12, 8, 4] and [12, 6, 4, 2].

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to partitions

Cf. A027750, A033630, A070824, A065205, A211111, A293813.

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1, n})[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
             b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i-1))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 16 2017

Table[d = Divisors[n]; Coefficient[Series[Product[1 + Boole[d[[k]] != 1 && d[[k]] != n] x^d[[k]], {k, Length[d]}], {x, 0, n}], x, n], {n, 0, 100}]

A293814(n) = { if(!n,return(1)); my(p=1); fordiv(n,d,if(d>1&&dAntti Karttunen, Dec 22 2017

;; Implements a simple backtracking algorithm:
(define (A293814 n) (if (<= n 1) (- 1 n) (let ((s (list 0))) (let fork ((r n) (divs (cdr (proper-divisors n)))) (cond ((zero? r) (set-car! s (+ 1 (car s)))) ((or (null? divs) (> (car divs) r)) #f) (else (begin (fork (- r (car divs)) (cdr divs)) (fork r (cdr divs)))))) (car s))))
(define (proper-divisors n) (reverse (cdr (reverse (divisors n)))))
(define (divisors n) (let loop ((k n) (divs (list))) (cond ((zero? k) divs) ((zero? (modulo n k)) (loop (- k 1) (cons k divs))) (else (loop (- k 1) divs)))))
;; Antti Karttunen, Dec 22 2017

a(n) = [x^n] Product_{d|n, 1 < d < n} (1 + x^d).

a(n) = A211111(n) - 1 for n > 1.

A294137 Number of compositions (ordered partitions) of n into nontrivial divisors of n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 5, 1, 2, 0, 50, 0, 2, 2, 55, 0, 185, 0, 243, 2, 2, 0, 8903, 1, 2, 19, 1219, 0, 48824, 0, 5271, 2, 2, 2, 1323569, 0, 2, 2, 369182, 0, 1659512, 0, 36636, 5111, 2, 0, 254187394, 1, 53535, 2, 223502, 0, 65005979, 2, 16774462, 2, 2, 0, 235105418684, 0, 2, 41386, 47350055, 2

Offset: 0

Ilya Gutkovskiy, Oct 23 2017

nonn

			a(8) = 5 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are nontrivial divisors {2, 4} therefore we have [4, 4], [4, 2, 2], [2, 4, 2], [2, 2, 4] and [2, 2, 2, 2].

Index entries for sequences related to compositions

Cf. A027750, A070824, A100346, A293813, A293814, A294138.

Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[d[[k]] != 1 && d[[k]] != n] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 65}]

a(n) = [x^n] 1/(1 - Sum_{d|n, 1 < d < n} x^d).

A327642 Number of partitions of n into divisors of n that are at most sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 1, 5, 4, 6, 1, 19, 1, 8, 6, 25, 1, 37, 1, 36, 8, 12, 1, 169, 6, 14, 10, 64, 1, 247, 1, 81, 12, 18, 8, 1072, 1, 20, 14, 405, 1, 512, 1, 144, 82, 24, 1, 2825, 8, 146, 18, 196, 1, 1000, 12, 743, 20, 30, 1, 19858, 1, 32, 112, 969, 14, 1728, 1, 324, 24, 1105

Offset: 0

Ilya Gutkovskiy, Sep 20 2019

a(n) > n if n is in A058080 Union {0}, and, a(n) <= n if n is in A007964; indeed, a(n) = n only for n = 1. - Bernard Schott, Sep 22 2019

			The divisors of 6 are 1, 2, 3, 6 and sqrt(6) = 2.449..., so the possible partitions are 1+1+1+1+1+1 = 1+1+1+1+2 = 1+1+2+2 = 2+2+2; thus a(6) = 4. - _Bernard Schott_, Sep 22 2019

David A. Corneth, Table of n, a(n) for n = 0..9999 (first 5001 terms from Robert Israel)

Cf. A018818, A210442, A211110, A293813.

[1] cat [#RestrictedPartitions(n,{d:d in Divisors(n)| d le Sqrt(n)}):n in [1..70]]; // Marius A. Burtea, Sep 20 2019

f:= proc(n) local x, t, S;
    S:= 1;
    for t in numtheory:-divisors(n) do
      if t^2 <= n then
        S:= series(S/(1-x^t),x,n+1);
      fi
    od;
    coeff(S,x,n);
end proc:
map(f, [$0..100]); # Robert Israel, Sep 22 2019

a[n_] := SeriesCoefficient[Product[1/(1 - Boole[d <= Sqrt[n]] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 70}]

a(n) = [x^n] Product_{d|n, d <= sqrt(n)} 1 / (1 - x^d).
a(p) = 1, where p is prime.
a(p*q) = q+1 if p <= q are primes. - Robert Israel, Sep 22 2019

A331979 Number of compositions (ordered partitions) of n into distinct nontrivial divisors of n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 894, 0, 0, 0, 24, 0, 6, 0, 0, 0, 0, 0, 894, 0, 0, 0, 0, 0, 30, 0, 120, 0, 0, 0, 19518, 0, 0, 0, 0, 0, 126, 0, 0, 0, 0, 0, 18558, 0, 0, 0, 0, 0, 6, 0, 864

Offset: 0

Ilya Gutkovskiy, Feb 03 2020

nonn

			a(12) = 6 because we have [6, 4, 2], [6, 2, 4], [4, 6, 2], [4, 2, 6], [2, 6, 4] and [2, 4, 6].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to compositions

Cf. A018818, A027750, A033630, A065205, A070824, A100346, A211111, A293813, A293814, A294137, A294138, A331927, A331928.

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1, n})[]]):
      b:= proc(m, i, p) option remember; `if`(m=0, p!, `if`(i<1, 0,
             b(m, i-1, p)+`if`(l[i]>m, 0, b(m-l[i], i-1, p+1))))
          end; forget(b):
      b(n, nops(l), 0)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 03 2020

a[n_] := If[n == 0, 1, Module[{b, l = Divisors[n] ~Complement~ {1, n}}, b[m_, i_, p_] := b[m, i, p] = If[m == 0, p!, If[i < 1, 0, b[m, i-1, p] + If[l[[i]] > m, 0, b[m - l[[i]], i-1, p+1]]]]; b[n, Length[l], 0]]];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

A357311 Number of partitions of n into divisors of n that are smaller than sqrt(n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 4, 1, 5, 1, 6, 1, 19, 1, 8, 6, 9, 1, 37, 1, 36, 8, 12, 1, 169, 1, 14, 10, 64, 1, 247, 1, 81, 12, 18, 8, 478, 1, 20, 14, 405, 1, 512, 1, 144, 82, 24, 1, 2825, 1, 146, 18, 196, 1, 1000, 12, 743, 20, 30, 1, 19858, 1, 32, 112, 289, 14, 1728, 1, 324, 24, 1105

Offset: 0

Ilya Gutkovskiy, Sep 23 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A018818, A210442, A293813, A327642, A357312.

a:= proc(n) option remember; uses numtheory; local b, l;
      l:= sort([select(x-> is(xm, 0, b(m-l[i], i))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Sep 23 2022

a[n_] := SeriesCoefficient[Product[1/(1 - Boole[d < Sqrt[n]] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 70}]

a(n) = [x^n] Product_{d|n, d < sqrt(n)} 1 / (1 - x^d).

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A293814 Number of partitions of n into distinct nontrivial divisors of n.

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A294137 Number of compositions (ordered partitions) of n into nontrivial divisors of n.

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A327642 Number of partitions of n into divisors of n that are at most sqrt(n).

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A331979 Number of compositions (ordered partitions) of n into distinct nontrivial divisors of n.

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A357311 Number of partitions of n into divisors of n that are smaller than sqrt(n).

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