A325801 -id:A325801 - cp's OEIS frontend

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

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).

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

Original entry on oeis.org

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)).

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.

A325835 Number of integer partitions of 2*n having one more distinct submultiset than distinct subset-sums.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 9, 10, 14, 22, 30, 33, 46, 52, 74, 107, 101, 123, 171, 182, 225

Offset: 0

Gus Wiseman, May 29 2019

The number of submultisets of a partition is the product of its multiplicities, each plus one. A subset-sum of an integer partition is the sum of some submultiset of its parts. These are partitions with one subset-sum which is the sum of two distinct submultisets, while all others are the sum of only one submultiset.

The Heinz numbers of these partitions are given by A325802.

			The a(2) = 1 through a(8) = 14 partitions:
  (211)  (321)   (422)    (532)     (633)      (743)       (844)
         (3111)  (431)    (541)     (642)      (752)       (853)
                 (41111)  (5221)    (651)      (761)       (862)
                          (5311)    (4332)     (7322)      (871)
                          (511111)  (5331)     (7331)      (5443)
                                    (6222)     (7421)      (7441)
                                    (6411)     (7511)      (7531)
                                    (33222)    (72221)     (8332)
                                    (6111111)  (74111)     (8521)
                                               (71111111)  (8611)
                                                           (82222)
                                                           (83311)
                                                           (85111)
                                                           (811111111)
For example, the partition (7,5,3,1) has submultisets (), (1), (3), (5), (7), (3,1), (5,1), (5,3), (7,1), (7,3), (7,5), (5,3,1), (7,3,1), (7,5,1), (7,5,3), (7,5,3,1), all of which have different sums except for (5,3) and (7,1), which both sum to 8, so (7,5,3,1) is counted under a(8).

Cf. A002033, A088880, A088881, A108917, A126796, A276024, A325799, A325800, A325801, A325802, A325828, A325830, A325833, A325836.

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==1+Length[Union[Total/@Subsets[#]]]&]],{n,0,20,2}]

A367105 Least positive integer with n more divisors than distinct subset-sums of prime indices.

Original entry on oeis.org

1, 12, 24, 48, 60, 192, 144, 120, 180, 336, 240, 630, 420, 360, 900, 1344, 960, 1008, 720, 840, 2340, 1980, 1260, 1440, 3120, 2640, 1680, 4032, 2880, 6840, 3600, 4620, 3780, 2520, 6480, 11700, 8820, 6300, 7200, 10560, 6720, 12240, 9360, 7920, 5040, 10920, 9240

Offset: 1

Gus Wiseman, Nov 09 2023

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.An integer n is a subset-sum (A299701, A304792) of a multiset y if there exists a submultiset of y with sum n.

			The divisors of 60 are {1,2,3,4,5,6,10,12,15,20,30,60}, and the distinct subset-sums of its prime indices {1,1,2,3} are {0,1,2,3,4,5,6,7}, so the difference is 12 - 8 = 4. Since 60 is the first number with this difference, we have a(4) = 60.
The terms together with their prime indices begin:
     1: {}
    12: {1,1,2}
    24: {1,1,1,2}
    48: {1,1,1,1,2}
    60: {1,1,2,3}
   120: {1,1,1,2,3}
   144: {1,1,1,1,2,2}
   180: {1,1,2,2,3}
   192: {1,1,1,1,1,1,2}
   240: {1,1,1,1,2,3}
   336: {1,1,1,1,2,4}
   360: {1,1,1,2,2,3}
   420: {1,1,2,3,4}
   630: {1,2,2,3,4}
   720: {1,1,1,1,2,2,3}
   840: {1,1,1,2,3,4}
   900: {1,1,2,2,3,3}
   960: {1,1,1,1,1,1,2,3}

The first part (divisors) is A000005.
The second part (subset-sums of prime indices) is A299701, positive A304793.
These are the positions of first appearances in the difference A325801.
The binary version is A367093, firsts of  A086971 - A366739.
A001222 counts prime factors (or prime indices), distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A220264, A304792, A365920, A366740, A367097.

nn=1000;
w=Table[DivisorSigma[0,n]-Length[Union[Total/@Subsets[prix[n]]]],{n,nn}];
spnm[y_]:=Max@@Select[Union[y],Function[i,Union[Select[y,#<=i&]]==Range[0,i]]];
Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

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

cp's OEIS Frontend

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

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

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A325835 Number of integer partitions of 2*n having one more distinct submultiset than distinct subset-sums.

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A367105 Least positive integer with n more divisors than distinct subset-sums of prime indices.

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