A065219
Number of subsets of proper divisors of numbers in A065218 summing to the number.
Original entry on oeis.org
0, 1, 2, 5, 7, 10, 34, 278, 751, 2157, 22208, 676327, 2225346, 23259536, 265050967, 39161483067, 70455119174849, 776384598617893, 133991542908557129, 21819590324155207874, 263763825614848727692, 12883245190231409112736, 661394651111310011564685
Offset: 1
Proper divisors of 12 are {1, 2, 3, 4, 6}. Two subsets of this sum to 12: {2, 4, 6} and {1, 2, 3, 6} - more than any smaller number, so 2 is in the sequence (and 12 is in A065218).
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Union@ FoldList[Max, Array[Block[{dd = Most@ Divisors@ #, c, cc}, cc = Array[c, Length@ dd]; Length@{ ToRules[Reduce[And @@ (0 <= # <= 1 &) /@ cc && dd.cc == #, cc, Integers]]}] &, 360]] (* Michael De Vlieger, Oct 01 2017, after Jean-François Alcover at A065205 *)
Initial 0 prepended and offset corrected by
Amiram Eldar, Oct 01 2017
A065205
Number of subsets of proper divisors of n that sum to n.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 7, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 34, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 1, 0, 6, 0, 0, 0, 25, 0, 0, 0, 1, 0, 23, 0, 0, 0, 0, 0, 21, 0, 0, 0, 2
Offset: 1
Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 19 2001
a(20) = 1 because {1, 4, 5, 10} is the only subset of proper divisors of 20 that sum to 20.
a(24) = 5 because there are five different subsets we can use to sum up to 24: {1, 2, 3, 4, 6, 8}, {1, 2, 3, 6, 12}, {1, 3, 8, 12}, {2, 4, 6, 12}, {4, 8, 12}.
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a065205 n = p (a027751_row n) n where
p _ 0 = 1
p [] _ = 0
p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Jan 21 2013
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a[n_] := (dd = Most[ Divisors[n] ]; cc = Array[c, Length[dd]]; Length[ {ToRules[ Reduce[ And @@ (0 <= # <= 1 &) /@ cc && dd . cc == n, cc, Integers]]}]); Table[ a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 23 2012 *)
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a(n,s,d)={s || (s=sigma(n)-n) || return; d||d=vecextract(divisors(n),"^-1"); while(d[#d]>n, s-=d[#d]; d=d[1..-2]); s<=n && return(s==n); if( n>d[#d], a(n-d[#d],s-d[#d],d[1..-2]), 1)+a(n,s-d[#d],d[1..-2])} \\ M. F. Hasler, May 11 2015
More terms and additional comments from
Jud McCranie, Oct 21 2001
Original entry on oeis.org
1, 6, 24, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640
Offset: 1
A359196
a(n) is the number of subsets of the divisors of n which sum to n+1.
Original entry on oeis.org
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 8, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 33, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 27, 1, 1, 1, 1, 1, 2, 1, 7, 1, 1, 1, 25, 1, 1, 1, 2, 1, 20, 1, 1, 1, 1, 1, 21, 1, 1, 1, 3
Offset: 1
a(1) = 0; a(2) = 1 since the divisors of 2, {1, 2} sum to 3;
a(18) = 2 since the divisors of 18, {1, 2, 3, 6, 9, 18}, have two subsets, {1, 18}, {1, 3, 6, 9} which sum to 19;
a(12) = 3 since the divisors of 12, {1, 2, 3, 4, 6, 12}, have three subsets, {1, 12}, {3, 4, 6}, {1, 2, 4, 6} which sum to 13;
a(162) = 4 since its divisors are {1, 2, 3, 6, 9, 18, 27, 54, 81, 162}, have four subsets, {1, 162}, {1, 27, 54, 81}, {1, 9, 18, 54, 81}, {1, 3, 6, 18, 54, 81} which sum to 163;
a(24) = 5 since its divisors {1, 2, 3, 4, 6, 8, 12, 24} have five subsets {1, 24}, {1, 4, 8, 12}, {2, 3, 8, 12}, {3, 4, 6, 12}, {1, 2, 4, 6, 12} which sum to 25; etc.
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a[n_] := Block[{c = k =1, d = Most@ Divisors@ n}, lgth = Length@ d; If[lgth < 18, c = 1 + Count[Total /@ Subsets@ d, n +1], While[k < 1 + 2^(lgth - 18), c += Count[Total /@ Subsets[d, All, {1 + (k -1)*2^18, k*2^18}], n +1]; k++]]; c]; Array[a, 100] (* or *)
a[n_] := Block[{d = Divisors@ n}, SeriesCoefficient[ Series[ Product[1 + x^d[[i]], {i, Length@ d}], {x, 0, n +1}], n +1]]; Array[a, 100]
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A359196(n) = if(!n, 0, if(sigma(n)<=n, 1, my(p=1); fordiv(n, d, p *= (1 + 'x^d)); polcoeff(p, 1+n))); \\ Antti Karttunen, Jan 20 2025
A096356
Smallest number which can be expressed as the sum of its proper divisors in exactly n ways.
Original entry on oeis.org
1, 6, 12, 30, 112, 24, 80, 36, 228, 150, 48, 156, 160, 126, 1242, 132, 5300, 1330, 448, 1326, 108, 96, 1288, 90, 918, 84, 1026, 750, 858, 16592, 744, 72, 910, 952, 60, 696, 896, 702, 690, 760, 6966, 12464, 192, 570, 400, 6642, 546, 594, 2178, 2420, 5424, 640
Offset: 0
a(2)=12 because 12 is the smallest number which can be expressed as the sum of its proper divisors in exactly 2 ways: 12=6+4+2 and 12=6+3+2+1.
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(* first *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Count[Plus @@@ Subsets[ Drop[ Divisors[n], -1]], n]; t = Table[0, {100}]; Do[ a = f[n]; If[a < 101 && t[[a]] == 0, t[[a]] = n; Print[a, " = ", n]], {n, 2, 16600}]; t (* Robert G. Wilson v, Aug 13 2004 *)
Showing 1-10 of 11 results.
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