A325694 -id:A325694 - cp's OEIS frontend

A325834 Number of integer partitions of n whose number of submultisets is less than or equal to n.

0, 0, 1, 1, 3, 3, 6, 7, 12, 14, 21, 21, 37, 43, 51, 56, 90, 98, 130, 143, 180, 200, 230, 249, 403, 454, 508, 555, 657, 706, 826, 889, 1295, 1406, 1568, 1690, 2194, 2396, 2603, 2841, 3387, 3672, 4024, 4344, 4693, 5079, 5489, 5840, 9731, 10424, 11336, 12093

Offset: 0

Gus Wiseman, May 29 2019

nonn

The number of submultisets of a partition is the product of its multiplicities, each plus one.

The Heinz numbers of these partitions are given by A325798.

			The a(2) = 1 through a(9) = 14 partitions:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (22)  (32)  (33)   (43)   (44)    (54)
            (31)  (41)  (42)   (52)   (53)    (63)
                        (51)   (61)   (62)    (72)
                        (222)  (322)  (71)    (81)
                        (411)  (331)  (332)   (333)
                               (511)  (422)   (432)
                                      (431)   (441)
                                      (521)   (522)
                                      (611)   (531)
                                      (2222)  (621)
                                      (5111)  (711)
                                              (3222)
                                              (6111)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A002033, A088880, A088881, A108917, A126796, A307699, A325694, A325792, A325798, A325828, A325830, A325831, A325832, A325833, A325836.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
      (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
    end:
a:= n-> add(b(n$2, k), k=0..n):
seq(a(n), n=0..55);  # Alois P. Heinz, Aug 17 2019

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])<=n&]],{n,0,30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, Function[w, b[w, Min[w, i - 1], Quotient[p, j + 1]]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := Sum[b[n, n, k], {k, 0, n}];
a /@ Range[0, 55] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = A000041(n) - A325831(n).

For n even, A325833(n) = a(n) - A325830(n/2); for n odd, A325833(n) = a(n).

A088881 If A056239(m) = n, then a(n) is the maximum value of A000005(m).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 20, 24, 30, 36, 42, 48, 60, 72, 84, 96, 112, 128, 144, 168, 192, 224, 256, 288, 336, 384, 432, 480, 540, 600, 672, 768, 864, 960, 1080, 1200, 1320, 1440, 1620, 1800, 1980, 2160, 2400, 2640, 2880, 3240, 3600, 3960, 4320, 4800, 5280

Offset: 0

Naohiro Nomoto, Nov 28 2003

Maximum number of submultisets among all integer partitions of n. - Gus Wiseman, Jun 30 2019

			The partition (3,2,1,1,1) has 16 submultisets, which is more than for any other partition of 8, so a(8) = 16. - _Gus Wiseman_, Jun 30 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A088880, A307699, A325694, A325798, A325828, A325830, A325831, A325832, A325833, A325834.

b:= proc(n, i) option remember; `if`(n=0 or i<2, n+1,
       max(seq((j+1)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n, n):
seq (a(n), n=0..100);  # Alois P. Heinz, Aug 09 2012

$RecursionLimit = 1000; b[n_, i_] :=  b[n, i] = If[n == 0 || i<2, n+1, Max[Table[ (j+1)*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table [a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 15 2015, after Alois P. Heinz *)
Table[Max@@(Times@@(1+Length/@Split[#])&)/@IntegerPartitions[n],{n,0,30}] (* Gus Wiseman, Jun 30 2019 *)

A325794 Number of divisors of n minus the sum of prime indices of n.

Original entry on oeis.org

1, 1, 0, 1, -1, 1, -2, 1, -1, 0, -3, 2, -4, -1, -1, 1, -5, 1, -6, 1, -2, -2, -7, 3, -3, -3, -2, 0, -8, 2, -9, 1, -3, -4, -3, 3, -10, -5, -4, 2, -11, 1, -12, -1, -1, -6, -13, 4, -5, -1, -5, -2, -14, 1, -4, 1, -6, -7, -15, 5, -16, -8, -2, 1, -5, 0, -17, -3, -7

Offset: 1

Gus Wiseman, May 23 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of positive terms are A325795.
Positions of nonnegative terms are A325796.
Positions of negative terms are A325797.
Positions of nonpositive terms are A325798.
Positions of 1's are A325792.
Positions of 0's are A325793.
Positions of -1's are A325694.
Cf. A000005, A002033, A056239, A112798, A299702, A325780, A325781.

Table[DivisorSigma[0,n]-Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A325794(n) = (numdiv(n)-A056239(n)); \\ Antti Karttunen, May 26 2019

a(n) = A000005(n) - A056239(n).

A325795 Numbers with more divisors than the sum of their prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 204, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 280, 288

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325781 in having 156.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n).

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Positions of positive terms in A325794.
Heinz numbers of the partitions counted by A325831.
Cf. A000005, A002033, A056239, A112798, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325796, A325797, A325798.

Select[Range[100],DivisorSigma[0,#]>Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

A325832 Number of integer partitions of n whose number of submultisets is greater than or equal to n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 8, 13, 16, 22, 35, 50, 58, 85, 120, 162, 199, 267, 347, 462, 592, 773, 1006, 1293, 1504, 1929, 2455, 3081, 3859, 4815, 5953, 7363, 8737, 10743, 13193, 16102, 19241, 23413, 28344, 34260, 40911, 49197, 58917, 70515, 84055, 100070, 118914

Offset: 0

Gus Wiseman, May 25 2019

nonn

The number of submultisets of a partition is the product of its multiplicities, each plus one.

The Heinz numbers of these partitions are given by A325796.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (21)   (31)    (221)    (321)     (421)      (431)
       (11)  (111)  (211)   (311)    (411)     (2221)     (521)
                    (1111)  (2111)   (2211)    (3211)     (3221)
                            (11111)  (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A002033, A098859, A108917, A126796, A325694, A325792, A325796, A325828, A325830, A325831, A325833, A325834, A325836.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
      (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
    end:
a:= n-> combinat[numbpart](n)-add(b(n$2, k), k=0..n-1):
seq(a(n), n=0..55);  # Alois P. Heinz, Aug 17 2019

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])>=n&]],{n,0,30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, Function [w, b[w, Min[w, i - 1], p/(j + 1)]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := PartitionsP[n] - Sum[b[n, n, k], {k, 0, n - 1}];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)

a(n) = A000041(n) - A325833(n).

For n even, a(n) = A325831(n) + A325830(n/2); for n odd, a(n) = A325831(n).

A325690 Number of length-3 integer partitions of n whose largest part is not the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 4, 3, 7, 6, 10, 9, 14, 13, 19, 17, 24, 23, 30, 28, 37, 35, 44, 42, 52, 50, 61, 58, 70, 68, 80, 77, 91, 88, 102, 99, 114, 111, 127, 123, 140, 137, 154, 150, 169, 165, 184, 180, 200, 196, 217, 212, 234, 230, 252, 247, 271, 266, 290, 285, 310

Offset: 0

Gus Wiseman, May 15 2019

nonn

Confirmed recurrence relation from Colin Barker for n <= 10000. - Fausto A. C. Cariboni, Feb 19 2022

			The a(3) = 1 through a(13) = 14 partitions (A = 10, B = 11):
  (111)  (221)  (222)  (322)  (332)  (333)  (433)  (443)  (444)   (544)
         (311)  (411)  (331)  (521)  (432)  (442)  (533)  (543)   (553)
                       (421)  (611)  (441)  (622)  (542)  (552)   (643)
                       (511)         (522)  (631)  (551)  (732)   (652)
                                     (531)  (721)  (632)  (741)   (661)
                                     (621)  (811)  (641)  (822)   (733)
                                     (711)         (722)  (831)   (742)
                                                   (731)  (921)   (751)
                                                   (821)  (A11)   (832)
                                                   (911)          (841)
                                                                  (922)
                                                                  (931)
                                                                  (A21)
                                                                  (B11)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

Column k = 3 of A325592.
Cf. A000041, A001399, A005044, A008642, A069905, A266223.
Cf. A325689, A325691, A325694.

Table[Length[Select[IntegerPartitions[n,{3}],#[[1]]!=#[[2]]+#[[3]]&]],{n,0,30}]

Conjectures from Colin Barker, May 15 2019: (Start)
G.f.: x^3*(1 + x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8.
(End)

A325796 Numbers with at least as many divisors as the sum of their prime indices.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 80, 84, 88, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240

Offset: 1

Gus Wiseman, May 23 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n).

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}

Positions of nonnegative terms in A325794.
Heinz numbers of the partitions counted by A325832.
Cf. A000005, A002033, A056239, A112798, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325795, A325797, A325798.

Select[Range[100],DivisorSigma[0,#]>=Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

A325797 Numbers with fewer divisors than the sum of their prime indices.

Original entry on oeis.org

5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97

Offset: 1

Gus Wiseman, May 23 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n).

			The sequence of terms together with their prime indices begins:
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  34: {1,7}
  35: {3,4}

Positions of negative terms in A325794.
Heinz numbers of the partitions counted by A325833.
Cf. A000005, A002033, A056239, A112798, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325795, A325796, A325798.

Select[Range[100],DivisorSigma[0,#]PrimePi[p]*k]]&]

A325689 Number of length-3 compositions of n such that no part is the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 0, 6, 4, 15, 12, 28, 24, 45, 40, 66, 60, 91, 84, 120, 112, 153, 144, 190, 180, 231, 220, 276, 264, 325, 312, 378, 364, 435, 420, 496, 480, 561, 544, 630, 612, 703, 684, 780, 760, 861, 840, 946, 924, 1035, 1012, 1128, 1104, 1225, 1200, 1326, 1300, 1431

Offset: 0

Gus Wiseman, May 15 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

Confirmed recurrence relation from Colin Barker for n <= 5000. - Fausto A. C. Cariboni, Feb 15 2022

			The a(3) = 1 through a(8) = 12 compositions (empty columns not shown):
  (111)  (113)  (114)  (115)  (116)
         (122)  (141)  (124)  (125)
         (131)  (222)  (133)  (152)
         (212)  (411)  (142)  (161)
         (221)         (151)  (215)
         (311)         (214)  (233)
                       (223)  (251)
                       (232)  (323)
                       (241)  (332)
                       (313)  (512)
                       (322)  (521)
                       (331)  (611)
                       (412)
                       (421)
                       (511)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..5000

Cf. A000079, A001399, A005044, A008642, A069905, A124278, A266223.

Cf. A325676, A325688, A325690, A325691, A325694.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],And@@Table[#[[i]]!=Total[Delete[#,i]],{i,3}]&]],{n,0,30}]

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 - x + 4*x^2) / ((1 - x)^3*(1 + x)^2) for n>5.
a(n) = -(5 + 3*(-1)^n - 2*n) * (n-2) / 4 for n>0.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
(End)

A325801 Number of divisors of n minus the number of distinct positive subset-sums of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, May 23 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n). A positive subset-sum of an integer partition is any sum of a nonempty submultiset of it.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's are A299702.
Positions of 1's are A325802.
Positions of positive integers are A299729.
Cf. A000005, A002033, A056239, A108917, A112798, A276024, A304793.
Cf. A325694, A325780, A325781, A325792, A325793, A325799.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
Table[DivisorSigma[0,n]-Length[Union[hwt/@Divisors[n]]],{n,100}]

A325801(n) = (numdiv(n) - A299701(n));
A299701(n) = { my(f = factor(n), pids = List([])); for(i=1,#f~, while(f[i,2], f[i,2]--; listput(pids,primepi(f[i,1])))); pids = Vec(pids); my(sv=vector(vecsum(pids))); for(b=1,(2^length(pids))-1,sv[sumbybits(pids,b)] = 1); 1+vecsum(sv); }; \\ Not really an optimal way to count these.
sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); }; \\ Antti Karttunen, May 26 2019

a(n) = A000005(n) - A299701(n).

cp's OEIS Frontend

A325834 Number of integer partitions of n whose number of submultisets is less than or equal to n.

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A088881 If A056239(m) = n, then a(n) is the maximum value of A000005(m).

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A325794 Number of divisors of n minus the sum of prime indices of n.

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A325795 Numbers with more divisors than the sum of their prime indices.

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A325832 Number of integer partitions of n whose number of submultisets is greater than or equal to n.

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A325690 Number of length-3 integer partitions of n whose largest part is not the sum of the other two.

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A325796 Numbers with at least as many divisors as the sum of their prime indices.

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A325797 Numbers with fewer divisors than the sum of their prime indices.

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