A325781 -id:A325781 - cp's OEIS frontend

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

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]]&]

A365923 Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 4, 0, 2, 0, 1, 0, 5, 1, 0, 3, 1, 1, 0, 8, 0, 3, 0, 3, 0, 1, 0, 10, 2, 1, 2, 2, 3, 1, 1, 0, 16, 0, 5, 0, 3, 0, 5, 0, 1, 0, 20, 2, 2, 4, 2, 6, 0, 4, 1, 1, 0, 31, 0, 6, 0, 8, 0, 5, 0, 5, 0, 1, 0, 39, 4, 4, 4, 1, 6, 6, 3, 2, 6, 1, 1, 0

Offset: 0

Gus Wiseman, Sep 24 2023

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

			The partition (4,2) has subset-sums {2,4,6} and non-subset-sums {1,3,5} so is counted under T(6,3).
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  1  1  1  0
   4  0  2  0  1  0
   5  1  0  3  1  1  0
   8  0  3  0  3  0  1  0
  10  2  1  2  2  3  1  1  0
  16  0  5  0  3  0  5  0  1  0
  20  2  2  4  2  6  0  4  1  1  0
  31  0  6  0  8  0  5  0  5  0  1  0
  39  4  4  4  1  6  6  3  2  6  1  1  0
  55  0 13  0  8  0 12  0  6  0  6  0  1  0
  71  5  8  7  3  5  3 16  3  6  0  6  1  1  0
Row n = 6 counts the following partitions:
  (321)     (411)  .  (51)   (33)  (6)  .
  (3111)              (42)
  (2211)              (222)
  (21111)
  (111111)

Row sums are A000041.
The rank statistic counted by this triangle is A325799.
The strict case is A365545, weighted row sums A365922.
The complement (positive subset-sum) is A365658.
Weighted row sums are A365918, for positive subset-sums A304792.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k, strict A365661.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A000009, A006827, A122768, A364272, A365919, A365921.

Table[Length[Select[IntegerPartitions[n], Length[Complement[Range[n], Total/@Subsets[#]]]==k&]], {n,0,10}, {k,0,n}]

A365920 Greatest non-subset-sum of the prime indices of n, or 0 if there is none.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 30 2023

nonn

This is the greatest element of {0,...,A056239(n)} that is not equal to A056239(d) for any divisor d|n, d>1. This definition is analogous to the Frobenius number of a numerical semigroup (see link), but it looks only at submultisets of a finite multiset, not all multisets of elements of a set.

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.

			The prime indices of 156 are {1,1,2,6}, with subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10, so a(156) = 5.

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

For binary indices instead of sums we have A063250.
Positions of first appearances > 2 are A065091.
Zeros are A325781, nonzeros A325798.
For prime indices instead of sums we have A339662, minimum A257993.
For least instead of greatest non-subset-sum we have A366128.
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A005117, A079068, A098743, A264401, A286469 or A286470, A339737, A339886.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Max@@Prepend[nmz[prix[n]],0],{n,100}]

A365545 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Sep 24 2023

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

Is column k = n - 7 given by A325695?

			Triangle begins:
  1
  1  0
  0  1  0
  1  0  1  0
  0  1  0  1  0
  0  0  2  0  1  0
  1  0  0  2  0  1  0
  1  0  0  0  3  0  1  0
  0  1  1  0  0  3  0  1  0
  0  0  3  0  0  0  4  0  1  0
  1  0  0  2  2  0  0  4  0  1  0
  1  0  0  0  5  0  0  0  5  0  1  0
  2  0  0  0  0  5  2  0  0  5  0  1  0
  2  0  1  0  0  0  8  0  0  0  6  0  1  0
  1  1  3  0  0  0  0  7  3  0  0  6  0  1  0
  2  0  4  0  1  0  0  0 12  0  0  0  7  0  1  0
  1  1  2  2  3  1  0  0  0 11  3  0  0  7  0  1  0
  2  0  3  0  7  0  1  0  0  0 16  0  0  0  8  0  1  0
  3  0  0  2  6  3  3  1  0  0  0 15  4  0  0  8  0  1  0
Row n = 12: counts the following partitions:
  (6,3,2,1)  .  .  .  .  (9,2,1)  (6,5,1)  .  .  (11,1)  .  (12)  .
  (5,4,2,1)              (8,3,1)  (6,4,2)        (10,2)
                         (7,4,1)                 (9,3)
                         (7,3,2)                 (8,4)
                         (5,4,3)                 (7,5)

Row sums are A000009, non-strict A000041.
The complement (positive subset-sums) is also A365545 with rows reversed.
Weighted row sums are A365922, non-strict A365918.
The non-strict version is A365923, complement A365658, rank stat A325799.
A046663 counts partitions without a subset summing to k, strict A365663.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k, strict A365661.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A284640, A304792, A364272, A365921.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Complement[Range[n], Total/@Subsets[#]]]==k&]],{n,0,10},{k,0,n}]

A365922 Number of non-subset-sums of strict integer partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 8, 11, 18, 25, 38, 51, 70, 93, 122, 159, 206, 263, 328, 420, 514, 645, 776, 967, 1154, 1413, 1686, 2042, 2414, 2890, 3394, 4062, 4732, 5598, 6494, 7652, 8836, 10329, 11884, 13833, 15830, 18376, 20936, 24131, 27476, 31547, 35780, 40966, 46292, 52737

Offset: 1

Gus Wiseman, Sep 23 2023

nonn

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

			The a(6) = 11 ways, showing each strict partition and its non-subset-sums:
    (6): 1,2,3,4,5
   (51): 2,3,4
   (42): 1,3,5
  (321):

The complement (positive subset-sums) is A284640, non-strict A276024.
Weighted row sums of A365545, non-strict A365923.
Row sums of A365663, non-strict A046663.
The non-strict version is A365918.
The zero-full complement (subset-sums) is A365925, non-strict A304792.
A000041 counts integer partitions, strict A000009.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k.
A365661 counts strict partitions w/ a subset summing to k.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A325799, A364272, A365658, A365919, A365921.

Table[Total[Length[Complement[Range[n], Total/@Subsets[#]]]& /@ Select[IntegerPartitions[n], UnsameQ@@#&]],{n,30}]

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]]&]

A325791 Number of necklace permutations of {1..n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.

Original entry on oeis.org

1, 1, 1, 2, 4, 20, 82, 252, 1074, 7912, 39552, 152680, 776094, 5550310, 30026848, 108376910

Offset: 0

Gus Wiseman, May 23 2019

A necklace permutation is a permutation that is either empty or whose first part is the minimum. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. The only circular subsequence of maximum length is the sequence itself, not any rotation of it. For example, the circular subsequences of (1,3,2) are: (), (1), (2), (3), (1,3), (2,1), (3,2), (1,3,2).

			The a(1) = 1 through a(5) = 20 permutations:
  (1)  (1,2)  (1,2,3)  (1,2,3,4)  (1,2,3,4,5)
              (1,3,2)  (1,3,2,4)  (1,2,3,5,4)
                       (1,4,2,3)  (1,2,4,3,5)
                       (1,4,3,2)  (1,2,4,5,3)
                                  (1,2,5,4,3)
                                  (1,3,2,5,4)
                                  (1,3,4,2,5)
                                  (1,3,4,5,2)
                                  (1,3,5,2,4)
                                  (1,3,5,4,2)
                                  (1,4,2,3,5)
                                  (1,4,2,5,3)
                                  (1,4,3,2,5)
                                  (1,4,5,2,3)
                                  (1,4,5,3,2)
                                  (1,5,2,3,4)
                                  (1,5,2,4,3)
                                  (1,5,3,2,4)
                                  (1,5,3,4,2)
                                  (1,5,4,3,2)

Cf. A002033, A008965, A032153, A103295, A126796, A304793, A325684, A325781, A325786, A325788, A325789, A325790.

subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Permutations[Range[n]],#=={}||First[#]==1&&Range[n*(n+1)/2]==Union[Total/@subalt[#]]&]],{n,0,5}]

a(11)-a(15) from Bert Dobbelaere, Nov 01 2020

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]]&]

cp's OEIS Frontend

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

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A365923 Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.

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A365920 Greatest non-subset-sum of the prime indices of n, or 0 if there is none.

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A365545 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.

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A365922 Number of non-subset-sums of strict integer partitions of n.

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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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A325791 Number of necklace permutations of {1..n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.

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