cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, May 23 2019

Keywords

Comments

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.

Examples

			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}

Crossrefs

Positions of -1's in A325794.
Cf. A000005, A056239, A112798, A325780, A325792, A325793, A325794, A325795, A325796, A325797, A325798, A325836.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, May 23 2019

Keywords

Comments

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).

Examples

			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}

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, May 23 2019

Keywords

Comments

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).

Examples

			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}

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, May 23 2019

Keywords

Comments

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).

Examples

			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}

Links

Crossrefs

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.

Programs

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

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

Views

Author

Gus Wiseman, May 25 2019

Keywords

Comments

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.

Examples

			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)

Links

Crossrefs

Cf. A002033, A098859, A126796, A325694, A325792, A325795, A325828, A325830, A325832, A325833, A325834, A325836.

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

a(n) = A000041(n) - A325834(n).
For n even, a(n) = A325832(n) - A325830(n/2); for n odd, a(n) = A325832(n).

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

Views

Author

Gus Wiseman, May 23 2019

Keywords

Comments

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).

Links

Crossrefs

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.

Programs

  • Mathematica
    Table[DivisorSigma[0,n]-Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]],{n,100}]
  • PARI
    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

Formula

a(n) = A000005(n) - A056239(n).

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

Views

Author

Gus Wiseman, May 23 2019

Keywords

Comments

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).

Examples

			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}

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, May 23 2019

Keywords

Comments

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).

Examples

			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}

Crossrefs

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.

Programs

  • Mathematica
    Select[Range[100],DivisorSigma[0,#]PrimePi[p]*k]]&]

A359420 Numbers that are both practical (A005153) and phi-practical (A260653).

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, 208, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 272, 280, 288
Offset: 1

Views

Author

Amiram Eldar, Dec 31 2022

Keywords

Comments

First differs from A325795 at n = 45, and from A325781 at n = 36.
Numbers k such that each number in the range 1..sigma(k) is a sum of distinct divisors of k, and each number in the range 1..k is a subsum of the multiset {phi(d) : d | k}.

Links

Crossrefs

Intersection of A005153 and A260653.
Cf. A000010 (phi), A000203 (sigma).
Cf. A325781, A325795.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {};
phiPracticalQ[n_] := If[n == 1, True, (lst = Sort@EulerPhi@Divisors[n]; ok = True; Do[If[lst[[m]] > Sum[lst[[l]], {l, 1, m - 1}] + 1, (ok = False; Break[])], {m, 1, Length[lst]}]; ok)]; (* Frank M Jackson's code at A260653 *)
Select[Range[300], pracQ[#] && phiPracticalQ[#] &]
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