A325694 -id:A325694 - cp's OEIS frontend

A325802 Numbers with one more divisor than distinct subset-sums of their prime indices.

12, 30, 40, 63, 70, 112, 154, 165, 198, 220, 273, 286, 325, 351, 352, 364, 442, 525, 550, 561, 595, 646, 675, 714, 741, 748, 765, 832, 850, 874, 918, 931, 952, 988, 1045, 1173, 1254, 1334, 1425, 1495, 1539, 1564, 1653, 1666, 1672, 1771, 1794, 1798, 1870, 1900

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. A subset-sum of an integer partition is any sum of a submultiset of it.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of the partitions counted by A325835.

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   30: {1,2,3}
   40: {1,1,1,3}
   63: {2,2,4}
   70: {1,3,4}
  112: {1,1,1,1,4}
  154: {1,4,5}
  165: {2,3,5}
  198: {1,2,2,5}
  220: {1,1,3,5}
  273: {2,4,6}
  286: {1,5,6}
  325: {3,3,6}
  351: {2,2,2,6}
  352: {1,1,1,1,1,5}
  364: {1,1,4,6}
  442: {1,6,7}
  525: {2,3,3,4}
  550: {1,3,3,5}
  561: {2,5,7}

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

Positions of 1's in A325801.
Cf. A000005, A056239, A108917, A112798, A276024, A299701, A299702.
Cf. A325694, A325780, A325781, A325799, A325800, A325835.

filter:= proc(n) local F,t,S,i;
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);
  S:= {0}:
  for t in F do
   S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);
  od;
  nops(S) = mul(t[2]+1,t=F)-1
end proc:
select(filter, [$1..2000]); # Robert Israel, Oct 30 2024

Select[Range[100],DivisorSigma[0,#]==1+Length[Union[hwt/@Divisors[#]]]&]

A000005(a(n)) = 1 + A299701(a(n)).

A307699 Numbers k such that there is no integer partition of k with exactly k-1 submultisets.

Original entry on oeis.org

0, 1, 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 38, 42, 44, 48, 50, 54, 60, 62, 66, 68, 72, 74, 80, 84, 86, 90, 92, 98, 102, 104, 108, 110, 114, 122, 126, 128, 132, 134, 138, 140, 146, 150, 152, 158, 164, 168, 170, 174, 180, 182, 186, 192, 194, 198, 200, 206

Offset: 1

Gus Wiseman, May 30 2019

nonn

After a(1) = 0, first differs from A229488 in lacking 56.
The number of submultisets of a partition is the product of its multiplicities, each plus one.
{a(n)-1} contains all odd numbers m = p*q*... such that gcd(p-1, q-1, ...) > 2. In particular, {a(n)-1} contains all powers of all primes > 3. Proof: If g is the greatest common divisor, then all factors of k are congruent to 1 modulo g, and thus all multiplicities of any valid multiset are divisible by g. However, the required sum is congruent to 2 modulo g, and so no such multiset can exist. - Charlie Neder, Jun 06 2019

			The sequence of positive terms together with their prime indices begins:
   1: {}
   2: {1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  14: {1,4}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  30: {1,2,3}
  32: {1,1,1,1,1}
  38: {1,8}
  42: {1,2,4}
  44: {1,1,5}
  48: {1,1,1,1,2}
  50: {1,3,3}
  54: {1,2,2,2}
  60: {1,1,2,3}
Partitions realizing the desired number of submultisets for each non-term are:
   3: (3)
   4: (22)
   5: (41)
   7: (511)
   9: (621)
  10: (4411)
  11: (71111)
  13: (9211)
  15: (9111111)
  16: (661111)
  17: (9521)
  19: (94411)
  21: (981111)
  22: (88111111)
  23: (32222222222)
  25: (99421)
  27: (3222222222222)
  28: (994411)
  29: (98222222)

Positions of zeros in A325836.

Cf. A002033, A088880, A088881, A098859, A108917, A126796, A276024, A325694, A325792, A325798, A325828, A325830, A325833, A325834, A325835.

Select[Range[50],Function[n,Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])==n-1&]=={}]]

More terms from Alois P. Heinz, May 30 2019

A325695 Number of length-3 strict integer partitions of n such that the largest part is not the sum of the other two.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 5, 5, 8, 7, 12, 11, 16, 15, 21, 20, 27, 25, 33, 32, 40, 38, 48, 46, 56, 54, 65, 63, 75, 72, 85, 83, 96, 93, 108, 105, 120, 117, 133, 130, 147, 143, 161, 158, 176, 172, 192, 188, 208, 204, 225, 221, 243, 238, 261, 257, 280, 275

Offset: 0

Gus Wiseman, May 15 2019

nonn

			The a(7) = 1 through a(15) = 12 partitions (A = 10, B = 11, C = 12):
  (421)  (521)  (432)  (631)  (542)  (543)  (643)   (653)   (654)
                (531)  (721)  (632)  (732)  (652)   (842)   (753)
                (621)         (641)  (741)  (742)   (851)   (762)
                              (731)  (831)  (751)   (932)   (843)
                              (821)  (921)  (832)   (941)   (852)
                                            (841)   (A31)   (861)
                                            (931)   (B21)   (942)
                                            (A21)           (951)
                                                            (A32)
                                                            (A41)
                                                            (B31)
                                                            (C21)

Cf. A000041, A001399, A005044, A008642, A069905, A124278.

Cf. A325686, A325690, A325691, A325694, A325696.

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

Conjectures from Colin Barker, May 15 2019: (Start)
G.f.: x^7*(1 + x + 2*x^2) / ((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>9.
(End)
a(n) = A325696(n)/6. - Alois P. Heinz, Jun 18 2020

A325800 Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.

Original entry on oeis.org

3, 10, 28, 66, 88, 156, 208, 306, 340, 408, 544, 570, 684, 760, 912, 966, 1216, 1242, 1288, 1380, 1656, 1840, 2208, 2436, 2610, 2900, 2944, 3132, 3248, 3480, 3906, 4092, 4176, 4340, 4640, 4650, 5022, 5208, 5456, 5568, 5580, 6200, 6696, 6944, 7326, 7424, 7440

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325793 in lacking 70.
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 subset-sum of an integer partition is any sum of a submultiset of it.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose sum is equal to their number of distinct subset-sums. The enumeration of these partitions by sum is given by A126796 interlaced with zeros.

			340 has prime indices {1,1,3,7} which sum to 12 and have 12 distinct subset-sums: {0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12}, so 340 is in the sequence.
The sequence of terms together with their prime indices begins:
     3: {2}
    10: {1,3}
    28: {1,1,4}
    66: {1,2,5}
    88: {1,1,1,5}
   156: {1,1,2,6}
   208: {1,1,1,1,6}
   306: {1,2,2,7}
   340: {1,1,3,7}
   408: {1,1,1,2,7}
   544: {1,1,1,1,1,7}
   570: {1,2,3,8}
   684: {1,1,2,2,8}
   760: {1,1,1,3,8}
   912: {1,1,1,1,2,8}
   966: {1,2,4,9}
  1216: {1,1,1,1,1,1,8}
  1242: {1,2,2,2,9}
  1288: {1,1,1,4,9}
  1380: {1,1,2,3,9}

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

Positions of 1's in A325799.
Includes A239885 except for 1.
Cf. A002033, A056239, A108917, A112798, A126796, A299701, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325801, A325802.

filter:= proc(n) local F,t,S,i,r;
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);
  S:= {0}:
  for t in F do
   S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);
  od;
  nops(S) = add(t[1]*t[2],t=F)
end proc:
select(filter, [$1..10000]); # Robert Israel, Oct 30 2024

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],hwt[#]==Length[Union[hwt/@Divisors[#]]]&]

A056239(a(n)) = A299701(a(n)) = A304793(a(n)) + 1.

A325987 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 1, 3, 0, 1, 1, 2, 1, 1, 0, 1, 0, 3, 0, 3, 0, 4, 0, 1, 0, 3, 0, 1, 1, 3, 1, 3, 0, 3, 2, 1, 0, 4, 0, 1, 1, 1, 0, 1, 0, 5, 0, 3, 0, 5, 0, 3, 0, 6, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 1, 4, 0

Offset: 0

Gus Wiseman, May 30 2019

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

			Triangle begins:
  1
  0 1
  0 1 1
  0 1 0 2
  0 1 1 1 1 1
  0 1 0 2 0 3 0 1
  0 1 1 3 0 1 1 2 1 1
  0 1 0 3 0 3 0 4 0 1 0 3
  0 1 1 3 1 3 0 3 2 1 0 4 0 1 1 1
  0 1 0 5 0 3 0 5 0 3 0 6 0 1 0 3 0 2 0 1
  0 1 1 4 0 5 0 7 2 1 1 4 0 1 2 5 0 3 0 2 1 0 0 2
Row n = 7 counts the following partitions (empty columns not shown):
  (7)  (43)  (322)  (421)      (31111)  (3211)
       (52)  (331)  (2221)              (22111)
       (61)  (511)  (4111)              (211111)
                    (1111111)

Alois P. Heinz, Rows n = 0..60, flattened

Row lengths are A088881.
Row sums are A000041.
Diagonal n = k is A325830 interspersed with zeros.
Diagonal n + 1 = k is A325828.
Diagonal n - 1 = k is A325836.
Column k = 3 appears to be A137719.
Cf. A000005, A000712, A002033, A005179, A088880, A108917, A126796.
Cf. A325694, A325792, A325793, A325831, A325832, A325833, A325834.

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==k&]],{n,0,10},{k,1,Max@@(Times@@(1+Length/@Split[#])&)/@IntegerPartitions[n]}]

Sum_{k=1..A088881(n)} k * T(n,k) = A000712(n). - Alois P. Heinz, Aug 17 2019

cp's OEIS Frontend

A325802 Numbers with one more divisor than distinct subset-sums of their prime indices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A307699 Numbers k such that there is no integer partition of k with exactly k-1 submultisets.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A325695 Number of length-3 strict integer partitions of n such that the largest part is not the sum of the other two.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A325800 Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A325987 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula