A325793 -id:A325793 - cp's OEIS frontend

A325694 Numbers with one fewer divisors than the sum of their prime indices.

5, 9, 14, 15, 44, 45, 50, 78, 104, 105, 110, 135, 196, 225, 272, 276, 342, 380, 405, 476, 572, 585, 608, 650, 693, 726, 735, 825, 888, 930, 968, 1125, 1215, 1218, 1240, 1472, 1476, 1482, 1518, 1566, 1610, 1624, 1976, 1995, 2024, 2090, 2210, 2256, 2565, 2618

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 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 A325836.

			The sequence of terms together with their prime indices begins:
     5: {3}
     9: {2,2}
    14: {1,4}
    15: {2,3}
    44: {1,1,5}
    45: {2,2,3}
    50: {1,3,3}
    78: {1,2,6}
   104: {1,1,1,6}
   105: {2,3,4}
   110: {1,3,5}
   135: {2,2,2,3}
   196: {1,1,4,4}
   225: {2,2,3,3}
   272: {1,1,1,1,7}
   276: {1,1,2,9}
   342: {1,2,2,8}
   380: {1,1,3,8}
   405: {2,2,2,2,3}
   476: {1,1,4,7}

Positions of -1's in A325794.

Cf. A000005, A056239, A112798, A325780, A325792, A325793, A325794, A325795, A325796, A325797, A325798, A325836.

Select[Range[1000],DivisorSigma[0,#]==Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]-1&]

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

Original entry on oeis.org

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]]&]

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]]&]

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 *)

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

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]]&]

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]]&]

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

A325694 Numbers with one fewer divisors than the sum of their prime indices.

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

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

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

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

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Maple

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

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