A325694 -id:A325694 - cp's OEIS frontend

A325792 Positive integers with as many proper divisors as the sum of their prime indices.

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 204, 234, 256, 260, 294, 308, 315, 350, 392, 416, 486, 500, 512, 690, 696, 798, 920, 1024, 1026, 1064, 1088, 1116, 1122, 1190, 1365, 1430, 1458, 1496, 1755, 1936, 1968, 2025, 2048, 2058, 2079

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325780 in having 204.

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 term 42 is in the sequence because it has 7 proper divisors (1, 2, 3, 6, 7, 14, 21) and its sum of prime indices is also 1 + 2 + 4 = 7.
The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    32: {1,1,1,1,1}
    42: {1,2,4}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   128: {1,1,1,1,1,1,1}
   162: {1,2,2,2,2}
   176: {1,1,1,1,5}
   204: {1,1,2,7}
   234: {1,2,2,6}
   256: {1,1,1,1,1,1,1,1}

Positions of 1's in A325794.
Heinz numbers of the partitions counted by A325828.
Cf. A000005, A002033, A056239, A112798, A299702, A304793.
Cf. A325694, A325780, A325781, A325793, A325795, A325796, A325797, A325798.

Select[Range[100],DivisorSigma[0,#]-1==Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

A325799 Sum of the prime indices of n minus the number of distinct positive subset-sums of the prime indices of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 2, 1, 4, 0, 5, 2, 2, 0, 6, 0, 7, 0, 3, 3, 8, 0, 4, 4, 3, 1, 9, 0, 10, 0, 4, 5, 4, 0, 11, 6, 5, 0, 12, 0, 13, 2, 2, 7, 14, 0, 6, 2, 6, 3, 15, 0, 5, 0, 7, 8, 16, 0, 17, 9, 4, 0, 6, 1, 18, 4, 8, 2, 19, 0, 20, 10, 3, 5, 6, 2, 21, 0, 4, 11

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.

			The prime indices of 21 are {2,4}, with positive subset-sums {2,4,6}, so a(21) = 6 - 3 = 3.

Positions of 1's are A325800.
Positions of nonzero terms are A325798.
Cf. A000005, A002033, A056239, A108917, A112798, A276024, A299702, A304793.
Cf. A325694, A325780, A325794, A325801, A325802.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
Table[hwt[n]-Length[Union[hwt/@Rest[Divisors[n]]]],{n,30}]

a(n) = A056239(n) - A304793(n).

A325798 Numbers with at most as many divisors as the sum of their prime indices.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 28, 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, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from the complement of A325781 in lacking 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:
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  10: {1,3}
  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}
  28: {1,1,4}
  29: {10}
  31: {11}

Positions of nonpositive terms in A325794.
Heinz numbers of the partitions counted by A325834.
Cf. A000005, A002033, A056239, A112798, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325795, A325796, A325797.

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

A325793 Positive integers whose number of divisors is equal to their sum of prime indices.

Original entry on oeis.org

3, 10, 28, 66, 70, 88, 208, 228, 306, 340, 364, 490, 495, 525, 544, 550, 675, 744, 870, 966, 1160, 1216, 1242, 1254, 1288, 1326, 1330, 1332, 1672, 1768, 1785, 1870, 2002, 2064, 2145, 2295, 2457, 2900, 2944, 3250, 3280, 3430, 3468, 3540, 3724, 4125, 4144, 4248

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 term 70 is in the sequence because it has 8 divisors {1, 2, 5, 7, 10, 14, 35, 70} and its sum of prime indices is also 1 + 3 + 4 = 8.
The sequence of terms together with their prime indices begins:
     3: {2}
    10: {1,3}
    28: {1,1,4}
    66: {1,2,5}
    70: {1,3,4}
    88: {1,1,1,5}
   208: {1,1,1,1,6}
   228: {1,1,2,8}
   306: {1,2,2,7}
   340: {1,1,3,7}
   364: {1,1,4,6}
   490: {1,3,4,4}
   495: {2,2,3,5}
   525: {2,3,3,4}
   544: {1,1,1,1,1,7}
   550: {1,3,3,5}
   675: {2,2,2,3,3}
   744: {1,1,1,2,11}
   870: {1,2,3,10}
   966: {1,2,4,9}

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 0's in A325794.
Contains A239885 except for 1.
Cf. A000005, A056239, A112798, A299702, A304793.
Cf. A325694, A325780, A325781, A325792, A325795, A325796, A325797, A325798.

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(numtheory:-pi(t[1])*t[2],t=F) = mul(t[2]+1,t=F)
end proc:
select(filter, [$1..10000]); # Robert Israel, Oct 16 2023

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

A325691 Number of length-3 integer partitions of n whose largest part is not greater than the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 8, 11, 10, 13, 12, 15, 14, 18, 16, 20, 19, 23, 21, 26, 24, 29, 27, 32, 30, 36, 33, 39, 37, 43, 40, 47, 44, 51, 48, 55, 52, 60, 56, 64, 61, 69, 65, 74, 70, 79, 75, 84, 80, 90, 85, 95, 91, 101, 96, 107, 102, 113

Offset: 0

Gus Wiseman, May 15 2019

Also the number of possible triples of edge-lengths of a triangle with perimeter n, where degenerate (self-intersecting) triangles are allowed.

The number of triples (a,b,c) for 1 <= a <= b <= c <= a+b and a+b+c = n. - Yuchun Ji, Oct 15 2020

			The a(3) = 1 through a(12) = 6 partitions:
  (111)  (211)  (221)  (222)  (322)  (332)  (333)  (433)  (443)  (444)
                       (321)  (331)  (422)  (432)  (442)  (533)  (543)
                                     (431)  (441)  (532)  (542)  (552)
                                                   (541)  (551)  (633)
                                                                 (642)
                                                                 (651)

Stefano Spezia, Table of n, a(n) for n = 0..10000
Colin Defant, Michael Joseph, Matthew Macauley, and Alex McDonough, Torsors and tilings from toric toggling, arXiv:2305.07627 [math.CO], 2023. See g.f. at p. 20.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1).

Cf. A001399, A005044 (nondegenerate triangles), A008642, A069905, A124278.

Cf. A325688, A325690, A325694.

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

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 + x - 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)
a(n) = A005044(n+3) - A000035(n+3). i.e., remove the only one triple (a=0,b,b) if n is even from the A005044 which is the number of triples (a,b,c) for 0 <= a <= b <= c <= a+b and a+b+c = n. - Yuchun Ji, Oct 15 2020
The above conjectured formulas are true. - Stefano Spezia, May 19 2023

A325828 Number of integer partitions of n having exactly n + 1 submultisets.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 12, 1, 3, 4, 21, 1, 14, 1, 18, 4, 3, 1, 116, 3, 3, 12, 25, 1, 40, 1, 271, 4, 3, 4, 325, 1, 3, 4, 295, 1, 56, 1, 36, 47, 3, 1, 3128, 4, 32, 4, 44, 1, 407, 4, 566, 4, 3, 1, 1598, 1, 3, 65, 10656, 5, 90, 1, 54, 4, 84, 1

Offset: 0

Gus Wiseman, May 25 2019

nonn

The Heinz numbers of these partitions are given by A325792.

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

			The 12 = 11 + 1 submultisets of the partition (4331) are: (), (1), (3), (4), (31), (33), (41), (43), (331), (431), (433), (4331), so (4331) is counted under a(11).
The a(5) = 3 through a(11) = 12 partitions:
  221    111111  421      3311      22221      1111111111  4322
  311            2221     11111111  51111                  4331
  11111          4111               111111111              4421
                 1111111                                   5411
                                                           6221
                                                           6311
                                                           7211
                                                           33311
                                                           44111
                                                           222221
                                                           611111
                                                           11111111111

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

Cf. A002033, A098859, A126796, A188431, A325694, A325792, A325793, A325830, A325831, A325832, 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-> b(n$2,n+1):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 17 2019

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])-1==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, r = Quotient[p, j + 1]; Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := b[n, n, n+1];
a /@ Range[0, 80] (* Jean-François Alcover, May 11 2021, after Alois P. Heinz *)

A325833 Number of integer partitions of n whose number of submultisets is less than n.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 7, 9, 14, 20, 21, 27, 43, 50, 56, 69, 98, 118, 143, 165, 200, 229, 249, 282, 454, 507, 555, 637, 706, 789, 889, 986, 1406, 1567, 1690, 1875, 2396, 2602, 2841, 3078, 3672, 3977, 4344, 4660, 5079, 5488, 5840, 6296, 10424, 11306

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 A325797.

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

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

Cf. A002033, A088880, A088881, A098859, A108917, A307699, A325694, A325792, A325797, A325828, A325830, A325831, A325832, 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-> 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[#])Jean-François Alcover, May 12 2021, after Alois P. Heinz *)

a(n) = A000041(n) - A325832(n).

For n even, a(n) = A325834(n) - A325830(n/2); for n odd, a(n) = A325834(n).

A325830 Number of integer partitions of 2n having exactly 2n submultisets.

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 10, 1, 21, 12, 15, 1, 121, 1, 20, 37, 309, 1, 319, 1, 309, 47, 33, 1, 3435, 30, 38, 405, 593, 1, 1574, 1, 11511, 80, 51, 77, 17552, 1, 56, 92, 13921, 1, 3060, 1, 1439, 2911, 69, 1, 234969, 56, 2044, 126, 1998, 1, 46488, 114, 36615, 137, 87, 1, 141906

Offset: 0

Gus Wiseman, May 25 2019

nonn

If n is odd, there are no integer partitions of n with exactly n submultisets, so this sequence gives only the even-indexed terms.
The number of submultisets of an integer partition is the product of its multiplicities, each plus one.
The Heinz numbers of these partitions are given by A325793.

			The 12 submultisets of the partition (7221) are (), (1), (2), (7), (21), (22), (71), (72), (221), (721), (722), (7221), so (7221) is counted under a(6).
The a(1) = 1 through a(8) = 21 partitions (A = 10, B = 11):
  (2)  (31)  (411)  (431)   (61111)  (4332)    (8111111)  (6532)
                    (521)            (4431)               (6541)
                    (5111)           (5322)               (7432)
                                     (5331)               (7531)
                                     (6411)               (7621)
                                     (7221)               (8431)
                                     (7311)               (8521)
                                     (8211)               (9421)
                                     (33222)              (A321)
                                     (711111)             (44431)
                                                          (53332)
                                                          (63331)
                                                          (64222)
                                                          (73222)
                                                          (76111)
                                                          (85111)
                                                          (92221)
                                                          (94111)
                                                          (A3111)
                                                          (B2111)
                                                          (91111111)

Alois P. Heinz, Table of n, a(n) for n = 0..700 (first 101 terms from Andrew Howroyd)

Cf. A002033, A098859, A108917, A126796, A237999, A325694, A325792, A325793, A325828, A325831, A325832, 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-> `if`(isprime(n), 1, b(2*n$3)):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 16 2019

Table[Length[Select[IntegerPartitions[2*n],Times@@(1+Length/@Split[#])==2*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, r = p/(j + 1);
     Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := If[PrimeQ[n], 1, b[2n, 2n, 2n]];
a /@ Range[0, 60] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)

a(n)={if(n<1, 0, my(v=vector(2*n+1, k, vector(2*n))); v[1][1]=1; for(k=1, 2*n, forstep(j=#v, k, -1, for(m=1, (j-1)\k, for(i=1, 2*n\(m+1), v[j][i*(m+1)] += v[j-m*k][i])))); v[#v][2*n])} \\ Andrew Howroyd, Aug 16 2019

a(p) = 1 for prime p. - Andrew Howroyd, Aug 16 2019

Terms a(31) and beyond from Andrew Howroyd, Aug 16 2019

A325836 Number of integer partitions of n having n - 1 different submultisets.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 0, 3, 0, 5, 2, 2, 0, 15, 0, 2, 3, 25, 0, 17, 0, 18, 3, 2, 0, 150, 0, 2, 13, 24, 0, 43, 0, 351, 3, 2, 2, 383, 0, 2, 3, 341, 0, 60, 0, 37, 51, 2, 0, 3733, 0, 31, 3, 42, 0, 460, 1, 633, 3, 2, 0, 1780, 0, 2, 68, 12460, 0, 87, 0, 55, 3

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 A325694.

			The a(3) = 1 through a(13) = 15 partitions (empty columns not shown):
  (3)  (22)  (32)  (322)  (432)   (3322)  (32222)  (4432)
             (41)  (331)  (531)   (4411)  (71111)  (5332)
                   (511)  (621)                    (5422)
                          (3222)                   (5521)
                          (6111)                   (6322)
                                                   (6331)
                                                   (6511)
                                                   (7411)
                                                   (8221)
                                                   (8311)
                                                   (9211)
                                                   (33322)
                                                   (55111)
                                                   (322222)
                                                   (811111)

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

Positions of zeros are A307699.

Cf. A002033, A088880, A088881, A108917, A325694, A325768, A325792, A325798, A325828, A325830, A325833, A325835.

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-> b(n$2,n-1):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 17 2019

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==n-1&]],{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, r = p/(j + 1);
     Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := b[n, n, n-1];
a /@ Range[0, 80] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)

A325831 Number of integer partitions of n whose number of submultisets is greater than n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 8, 10, 16, 21, 35, 40, 58, 84, 120, 141, 199, 255, 347, 447, 592, 772, 1006, 1172, 1504, 1928, 2455, 3061, 3859, 4778, 5953, 7054, 8737, 10742, 13193, 15783, 19241, 23412, 28344, 33951, 40911, 49150, 58917, 70482, 84055, 100069, 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 A325795.

			The a(1) = 1 through a(8) = 10 partitions:
  (1)  (11)  (21)   (211)   (221)    (321)     (421)      (3221)
             (111)  (1111)  (311)    (2211)    (2221)     (3311)
                            (2111)   (3111)    (3211)     (4211)
                            (11111)  (21111)   (4111)     (22211)
                                     (111111)  (22111)    (32111)
                                               (31111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)

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

Cf. A002033, A098859, A126796, A325694, A325792, A325795, A325828, A325830, A325832, 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):
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}];
a /@ Range[0, 55] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

a(n) = A000041(n) - A325834(n).

For n even, a(n) = A325832(n) - A325830(n/2); for n odd, a(n) = A325832(n).

cp's OEIS Frontend

A325792 Positive integers with as many proper divisors as the sum of their prime indices.

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A325799 Sum of the prime indices of n minus the number of distinct positive subset-sums of the prime indices of n.

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A325798 Numbers with at most as many divisors as the sum of their prime indices.

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A325793 Positive integers whose number of divisors is equal to their sum of prime indices.

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A325691 Number of length-3 integer partitions of n whose largest part is not greater than the sum of the other two.

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A325828 Number of integer partitions of n having exactly n + 1 submultisets.

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A325833 Number of integer partitions of n whose number of submultisets is less than n.

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A325830 Number of integer partitions of 2*n having exactly 2*n submultisets.

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A325830 Number of integer partitions of 2n having exactly 2n submultisets.