A211111 -id:A211111 - cp's OEIS frontend

A033630 Number of partitions of n into distinct divisors of n.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 8, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 35, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 2, 1, 7, 1, 1, 1, 26, 1, 1, 1, 2, 1, 24, 1, 1, 1, 1, 1, 22, 1, 1, 1, 3

Offset: 0

Marc LeBrun

nonn

			a(12) = 3 because we have the partitions [12], [6, 4, 2], and [6, 3, 2, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (1000 terms from T. D. Noe)
Noah Lebowitz-Lockard and Joseph Vandehey, On the number of partitions of a number into distinct divisors, arXiv:2402.08119 [math.NT], 2024. See p. 2.

Cf. A018818, A065205.
Cf. A083206. - Reinhard Zumkeller, Jul 19 2010
Cf. A000009, A005153.
Cf. A211111, A027750.
Cf. A225245.
Cf. A000396, A005100, A005835, A006037.

a033630 0 = 1
a033630 n = p (a027750_row n) n where
   p _  0 = 1
   p [] _ = 0
   p (d:ds) m = if d > m then 0 else p ds (m - d) + p ds m
-- Reinhard Zumkeller, Feb 23 2014, Apr 04 2012, Oct 27 2011

with(numtheory): a:=proc(n) local div, g, gser: div:=divisors(n): g:=product(1+x^div[j],j=1..tau(n)): gser:=series(g,x=0,105): coeff(gser,x^n): end: seq(a(n),n=1..100); # Emeric Deutsch, Mar 30 2006
# second Maple program:
with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(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, Feb 05 2014

A033630 = Table[SeriesCoefficient[Series[Times@@((1 + z^#) & /@ Divisors[n]), {z, 0, n}], n ], {n, 512}] (* Wouter Meeussen *)
A033630[n_] := f[n, n, 1]; f[n_, m_, k_] := f[n, m, k] = If[k <= m, f[n, m, k + 1] + f[n, m - k, k + 1] * Boole[Mod[n, k] == 0], Boole[m == 0]]; Array[A033630, 101, 0] (* Jean-François Alcover, Jul 29 2015, after Reinhard Zumkeller *)

a(n) = A065205(n) + 1.
a(A005100(n)) = 1; a(A005835(n)) > 1. - Reinhard Zumkeller, Mar 02 2007
a(n) = f(n, n, 1) with f(n, m, k) = if k <= m then f(n, m, k + 1) + f(n, m - k, k + 1)*0^(n mod k) else 0^m. - Reinhard Zumkeller, Dec 11 2009
a(n) = [x^n] Product_{d|n} (1 + x^d). - Ilya Gutkovskiy, Jul 26 2017
a(n) = 1 if n is deficient (A005100) or weird (A006037). a(n) = 2 if n is perfect (A000396). - Alonso del Arte, Sep 24 2017

More terms from Reinhard Zumkeller, Apr 21 2003

A211110 Number of partitions of n into divisors > 1 of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 12, 1, 3, 3, 10, 1, 15, 1, 16, 3, 3, 1, 80, 2, 3, 5, 20, 1, 94, 1, 36, 3, 3, 3, 280, 1, 3, 3, 158, 1, 154, 1, 28, 25, 3, 1, 1076, 2, 29, 3, 32, 1, 255, 3, 262, 3, 3, 1, 7026, 1, 3, 32, 202, 3, 321, 1, 40, 3, 302, 1, 12072, 1

Offset: 0

Reinhard Zumkeller, Apr 01 2012

nonn

a(A000040(n)) = 1; a(A002808(n)) > 1;
a(A001248(n)) = 2; a(A080257(n)) > 2;
a(A006881(n)) = 3; a(A033942(n)) > 3.

			a(10) = #{10, 5+5, 2+2+2+2+2} = 3;
a(11) = #{11} = 1;
a(12) = #{12, 6+6, 6+4+2, 6+3+3, 6+2+2+2, 4+4+4, 4+4+2+2, 4+3+3+2, 4+2+2+2+2, 3+3+3+3, 3+3+2+2+2, 6x2} = 12;
a(13) = #{13} = 1;
a(14) = #{14, 7+7, 2+2+2+2+2+2+2} = 3;
a(15) = #{15, 5+5+5, 3+3+3+3+3} = 3.

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

Cf. A211111, A018818, A027750.

Cf. A210442.

a211110 n = p (tail $ a027750_row n) n where
   p _      0 = 1
   p []     _ = 0
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1})[]]):
      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))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..100); # Alois P. Heinz, Feb 05 2014

a[n_] := Module[{b, l}, l = Rest[Divisors[n]]; b[m_, i_] := b[m, i] = If[m==0, 1, If[i<1, 0, b[m, i-1] + If[l[[i]]>m, 0, b[m-l[[i]], i]]]]; b[n, Length[l]]]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

isokp(p, n) = {for (k=1, #p, if ((p[k]==1) || (n % p[k]), return (0));); return (1);}
a(n) = {my(nb = 0); forpart(p=n, if (isokp(p,n), nb++)); nb;} \\ Michel Marcus, Jun 30 2015

A136446 Numbers n such that some subset of the numbers { 1 < d < n : d divides n } adds up to n.

Original entry on oeis.org

12, 18, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 90, 96, 100, 102, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246

Offset: 1

Joerg Arndt, Apr 06 2008

nonn

This is a subset of the pseudoperfect numbers A005835 and thus non-deficient (A023196), but in view of the definition actually abundant numbers (A005101). Sequence A122036 lists odd abundant numbers (A005231) which are not in this sequence. So far, 351351 is the only one we know. (As of today, no odd weird (A006037: abundant but not pseudoperfect) number is known.) - M. F. Hasler, Apr 13 2008
This sequence contains infinitely many odd elements: any proper multiple of any pseudoperfect number is in the sequence, so odd proper multiples of odd pseudoperfect numbers are in the sequence. The first such is 2835 = 3 * 945 (which is in the b-file). - Franklin T. Adams-Watters, Jun 18 2009
A211111(a(n)) > 1. - Reinhard Zumkeller, Apr 04 2012

Mladen Vassilev, Two theorems concerning divisors, Bull. Number Theory Related Topics 12 (1988), pp. 10-19.

M. F. Hasler, Table of n, a(n) for n = 1..24491 (confirmed by _R. J. Mathar_, Mar 20 2011).

See A005835 (allowing for divisor 1).

Cf. A122036 = A005231 \ A136446.

a136446 n = a136446_list !! (n-1)
a136446_list = map (+ 1) $ findIndices (> 1) a211111_list
-- Reinhard Zumkeller, Apr 04 2012

isA136446a := proc(s,n) if n in s then return true; elif add(i,i=s) < n then return false; elif nops(s) = 1 then is(op(1,s)=n) ; else sl := sort(convert(s,list),`>`) ; for i from 1 to nops(sl) do m := op(i,sl) ; if n -m = 0 then return true; end if ; if n-m > 0 then sr := [op(i+1..nops(sl),sl)] ; if procname(convert(sr,set),n-m) then return true; end if; end if; end do; return false; end if; end proc:
isA136446 := proc(n) isA136446a( numtheory[divisors](n) minus {1,n},n) ; end proc:
for n from 1 to 400 do if isA136446(n) then printf("%d,",n) ; end if; end do ; # R. J. Mathar, Mar 20 2011

okQ[n_] := Module[{d}, If[PrimeQ[n], False, d = Most[Rest[Divisors[n]]]; MemberQ[Plus @@@ Subsets[d], n]]]; Select[Range[2, 246], okQ]
(* T. D. Noe, Jul 24 2012 *)

N=72 \\ up to this value
vv=vector(N);
{ for(n=2, N,
if ( isprime(n), next() );
d=divisors(n);
d=vector(#d-2,j,d[j+1]); \\ not n, not 1
for (k=1, (1<<#d)-1, \\ all subsets
t=vecextract(d, k);
if ( n==sum(j=1,#t,t[j]),
vv[n] += 1;););); }
for (j=1, #vv, if (vv[j]>0, print1(j,", "))) \\ A005835 (after correction)

is_A136446(n,d=divisors(n))={#d>2 && is_A005835(n,d[2..-2])} \\ Replaced old code not conforming to current PARI syntax. - M. F. Hasler, Jul 28 2016
for( n=1,10^4, is_A136446(n) && print1(n", ")) \\ M. F. Hasler, Apr 13 2008

def isa(s, n): # After R. J. Mathar's Maple code
    if n in s: return True
    if sum(s) < n: return False
    if len(s) == 1: return s[0] == n
    for i in srange(len(s)-1,-1,-1) :
        d = n - s[i]
        if d == 0: return True
        if d >  0:
            if isa(s[i+1:], d): return True
    return False
isA136446 = lambda n : isa(divisors(n)[1:-1], n)
[n for n in (1..246) if isA136446(n)]
# Peter Luschny, Jul 23 2012

More terms from M. F. Hasler, Apr 13 2008

A293814 Number of 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, 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.

A331927 Number of compositions (ordered partitions) of n into distinct divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 31, 1, 25, 1, 1, 1, 895, 1, 1, 1, 121, 1, 151, 1, 1, 1, 1, 1, 1135, 1, 1, 1, 865, 1, 31, 1, 1, 1, 1, 1, 11935, 1, 1, 1, 1, 1, 151, 1, 841, 1, 1, 1, 129439, 1, 1, 1, 1, 1, 127, 1, 1, 1, 1

Offset: 0

Ilya Gutkovskiy, Feb 01 2020

nonn

			a(6) = 7 because we have [6], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].

Index entries for sequences related to compositions

Cf. A018818, A027750, A033630, A065205, A100346, A210442, A211111, A294138, A331928.

a(n)={if(n==0, 1, my(v=divisors(n)); subst(serlaplace(polcoef(prod(i=1, #v, 1 + y*x^v[i] + O(x*x^n)), n)), y, 1))} \\ Andrew Howroyd, Feb 01 2020

a(n) = A331928(n) + 1 for n > 0.

A331928 Number of compositions (ordered partitions) of n into distinct proper divisors of n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 30, 0, 24, 0, 0, 0, 894, 0, 0, 0, 120, 0, 150, 0, 0, 0, 0, 0, 1134, 0, 0, 0, 864, 0, 30, 0, 0, 0, 0, 0, 11934, 0, 0, 0, 0, 0, 150, 0, 840, 0, 0, 0, 129438, 0, 0, 0, 0, 0, 126, 0, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Feb 01 2020

nonn

			a(6) = 6 because we have [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].

Index entries for sequences related to compositions

Cf. A018818, A027751, A033630, A065205, A100346, A210442, A211111, A294138, A331927.

a(n)={if(n==0, 1, my(v=divisors(n)); subst(serlaplace((0*y) + polcoef(prod(i=1, #v-1, 1 + y*x^v[i] + O(x*x^n)), n)), y, 1))} \\ Andrew Howroyd, Feb 01 2020

a(n) = A331927(n) - 1 for n > 0.

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 *)

cp's OEIS Frontend

A033630 Number of partitions of n into distinct divisors of n.

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A211110 Number of partitions of n into divisors > 1 of n.

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A136446 Numbers n such that some subset of the numbers { 1 < d < n : d divides n } adds up to n.

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

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A331927 Number of compositions (ordered partitions) of n into distinct divisors of n.

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A331928 Number of compositions (ordered partitions) of n into distinct proper divisors of n.

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

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