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 1-10 of 14 results. Next

A361856 Positive integers whose prime indices satisfy (maximum) = 2*(median).

Original entry on oeis.org

12, 24, 42, 48, 60, 63, 72, 96, 126, 130, 140, 144, 189, 192, 195, 252, 266, 288, 308, 325, 330, 360, 378, 384, 399, 420, 432, 495, 546, 567, 572, 576, 588, 600, 630, 638, 650, 665, 756, 768, 819, 864, 882, 884, 931, 945, 957, 962, 975, 1122, 1134, 1152, 1190
Offset: 1

Views

Author

Gus Wiseman, Apr 02 2023

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.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
These are Heinz numbers of partitions satisfying (maximum) = 2*(median).

Examples

			The terms together with their prime indices begin:
    12: {1,1,2}
    24: {1,1,1,2}
    42: {1,2,4}
    48: {1,1,1,1,2}
    60: {1,1,2,3}
    63: {2,2,4}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
   126: {1,2,2,4}
   130: {1,3,6}
   140: {1,1,3,4}
   144: {1,1,1,1,2,2}
The prime indices of 126 are {1,2,2,4}, with maximum 4 and median 2, so 126 is in the sequence.
The prime indices of 308 are {1,1,4,5}, with maximum 5 and median 5/2, so 308 is in the sequence.

Crossrefs

The LHS (greatest prime index) is A061395.
The RHS (twice median) is A360005, distinct A360457.
These partitions are counted by A361849.
For mean instead of median we have A361855, counted by A361853.
For minimum instead of median we have A361908, counted by A118096.
For length instead of median we have A361909, counted by A237753.
A000975 counts subsets with integer median.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
Cf. A027193, A067801, A111907, A361205, A361848, A361858.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@prix[#]==2*Median[prix[#]]&]

Formula

A061395(a(n)) = 2*A360005(a(n)).

A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 1, 2, 4, 0, 10, 0, 8, 16, 10, 0, 31, 0, 44, 44, 20, 0, 92, 50, 28, 98, 154, 0, 266, 0, 154, 194, 48, 434, 712, 0, 60, 348, 910, 0, 1198, 0, 1120, 2138, 88, 0, 2428, 1300, 1680, 912, 2506, 0, 4808, 4800, 5968, 1372, 140, 0, 14820, 0, 160
Offset: 1

Views

Author

Gus Wiseman, Mar 29 2023

Keywords

Comments

Also partitions satisfying (maximum) = 2*(mean).
These are partitions whose diagram has the same size as its complement (see example).

Examples

			The a(6) = 2 through a(12) = 10 partitions:
  (411)   .  (4211)  (621)     (5221)   .  (822)
  (3111)             (321111)  (5311)      (831)
                               (42211)     (6222)
                               (43111)     (6321)
                                           (6411)
                                           (422211)
                                           (432111)
                                           (441111)
                                           (32211111)
                                           (33111111)
The partition y = (6,4,1,1) has diagram:
  o o o o o o
  o o o o . .
  o . . . . .
  o . . . . .
Since the partition and its complement (shown in dots) have the same size, y is counted under a(12).

Crossrefs

For minimum instead of mean we have A118096.
For length instead of mean we have A237753.
For median instead of mean we have A361849, ranks A361856.
This is the equal case of A361851, unequal case A361852.
The strict case is A361854.
These partitions have ranks A361855.
This is the equal case of A361906, unequal case A361907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
A268192 counts partitions by complement size, ranks A326844.
Cf. A111907, A116608, A188814, A237755, A237824, A237984, A240219, A326849, A327482, A349156, A359894.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#==2n&]],{n,30}]

A361851 Number of integer partitions of n such that (length) * (maximum) <= 2*n.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 18, 23, 31, 37, 51, 58, 75, 96, 116, 126, 184, 193, 253, 307, 346, 402, 511, 615, 678, 792, 1045, 1088, 1386, 1419, 1826, 2181, 2293, 2779, 3568, 3659, 3984, 4867, 5885, 6407, 7732, 8124, 9400, 11683, 13025, 13269, 16216, 17774, 22016
Offset: 1

Views

Author

Gus Wiseman, Mar 28 2023

Keywords

Comments

Also partitions such that (maximum) <= 2*(mean).
These are partitions whose complement (see example) has size <= n.

Examples

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (411)     (421)
                                     (2211)    (2221)
                                     (3111)    (3211)
                                     (21111)   (22111)
                                     (111111)  (211111)
                                               (1111111)
The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 <= 2*7, so y is counted under a(7).
The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 is not <= 2*9, so y is not counted under a(9).
The partition y = (3,2,1,1) has diagram:
  o o o
  o o .
  o . .
  o . .
with complement of size 5, and 5 <= 7, so y is counted under a(7).

Crossrefs

For length instead of mean we have A237755.
For minimum instead of mean we have A237824.
For median instead of mean we have A361848.
The equal case for median is A361849, ranks A361856.
The unequal case is A361852, median A361858.
The equal case is A361853, ranks A361855.
Reversing the inequality gives A361906, unequal case A361907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
Cf. A111907, A237984, A240219, A324521, A324562, A327482, A349156, A360068, A360071, A360241, A361394, A361859.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#<=2n&]],{n,30}]

A361908 Positive integers > 1 whose prime indices satisfy (maximum) = 2*(minimum).

Original entry on oeis.org

6, 12, 18, 21, 24, 36, 48, 54, 63, 65, 72, 96, 105, 108, 133, 144, 147, 162, 189, 192, 216, 288, 315, 319, 324, 325, 384, 432, 441, 455, 481, 486, 525, 567, 576, 648, 715, 731, 735, 768, 845, 864, 931, 945, 972, 1007, 1029, 1152, 1296, 1323, 1403, 1458, 1463
Offset: 1

Views

Author

Gus Wiseman, Apr 05 2023

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.

Examples

			The terms together with their prime indices begin:
     6: {1,2}
    12: {1,1,2}
    18: {1,2,2}
    21: {2,4}
    24: {1,1,1,2}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    63: {2,2,4}
    65: {3,6}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}

Links

Crossrefs

The RHS is 2*A055396 (twice minimum).
The LHS is A061395 (greatest prime index).
Partitions of this type are counted by A118096.
For mean instead of minimum we have A361855, counted by A361853.
For median instead of minimum we have A361856, counted by A361849.
For length instead of minimum we have A361909, counted by A237753.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors with multiplicity.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A237820, A237821, A361858.

Programs

  • Maple
    filter:= proc(n) local F,b;
  if n::even then b:= padic:-ordp(n,3);
     if b = 0 then return false else return n = 2^padic:-ordp(n,2) * 3^b fi
  fi;
  F:= ifactors(n)[2][..,1];
  nops(F) >= 2 and numtheory:-pi(max(F)) = 2*numtheory:-pi(min(F))
end proc:
select(filter, [$1..2000]); # Robert Israel, Mar 11 2025
  • Mathematica
    Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]==2*PrimePi[FactorInteger[#][[1,1]]]&]

A361906 Number of integer partitions of n such that (length) * (maximum) >= 2*n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3, 5, 9, 15, 19, 36, 43, 68, 96, 125, 171, 232, 297, 418, 529, 676, 853, 1156, 1393, 1786, 2316, 2827, 3477, 4484, 5423, 6677, 8156, 10065, 12538, 15121, 17978, 22091, 26666, 32363, 38176, 46640, 55137, 66895, 79589, 92621, 111485, 133485
Offset: 1

Views

Author

Gus Wiseman, Mar 29 2023

Keywords

Comments

Also partitions such that (maximum) >= 2*(mean).
These are partitions whose complement (see example) has size >= n.

Examples

			The a(6) = 2 through a(10) = 15 partitions:
  (411)   (511)    (611)     (621)      (721)
  (3111)  (4111)   (4211)    (711)      (811)
          (31111)  (5111)    (5211)     (5221)
                   (41111)   (6111)     (5311)
                   (311111)  (42111)    (6211)
                             (51111)    (7111)
                             (321111)   (42211)
                             (411111)   (43111)
                             (3111111)  (52111)
                                        (61111)
                                        (421111)
                                        (511111)
                                        (3211111)
                                        (4111111)
                                        (31111111)
The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 >= 2*8, so y is counted under a(8).
The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 is not >= 2*7, so y is not counted under a(7).
The partition y = (3,2,1,1) has diagram:
  o o o
  o o .
  o . .
  o . .
with complement (shown in dots) of size 5, and 5 is not >= 7, so y is not counted under a(7).

Crossrefs

For length instead of mean we have A237752, reverse A237755.
For minimum instead of mean we have A237821, reverse A237824.
For median instead of mean we have A361859, reverse A361848.
The unequal case is A361907.
The complement is counted by A361852.
The equal case is A361853, ranks A361855.
Reversing the inequality gives A361851.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A268192 counts partitions by complement size, ranks A326844.
Cf. A027193, A111907, A116608, A237984, A324521, A327482, A349156, A360068.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#>=2n&]],{n,30}]

A361852 Number of integer partitions of n such that (length) * (maximum) < 2n.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 17, 21, 27, 37, 41, 58, 67, 80, 106, 126, 153, 193, 209, 263, 326, 402, 419, 565, 650, 694, 891, 1088, 1120, 1419, 1672, 1987, 2245, 2345, 2856, 3659, 3924, 4519, 4975, 6407, 6534, 8124, 8280, 9545, 12937, 13269, 13788, 16474, 20336
Offset: 1

Views

Author

Gus Wiseman, Mar 29 2023

Keywords

Comments

Also partitions such that (maximum) < 2*(mean).

Examples

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (2211)    (421)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (22111)
                                               (211111)
                                               (1111111)
For example, the partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 < 2*7, so y is counted under a(7).

Crossrefs

For length instead of mean we have A237754.
Allowing equality gives A237755, for median A361848.
For equal median we have A361849, ranks A361856.
The equal version is A361853, ranks A361855.
For median instead of mean we have A361858.
The complement is counted by A361906.
Reversing the inequality gives A361907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
Cf. A027193, A111907, A116608, A237824, A237984, A324517, A327482, A349156, A360068, A360071, A361394.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#<2n&]],{n,30}]

A361867 Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

Original entry on oeis.org

20, 28, 40, 44, 52, 56, 66, 68, 76, 78, 80, 84, 88, 92, 99, 102, 104, 112, 114, 116, 117, 120, 124, 132, 136, 138, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 190, 198, 200, 204, 207, 208, 212, 220, 222, 224, 228, 230, 232, 234
Offset: 1

Views

Author

Gus Wiseman, Apr 05 2023

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.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 > 2*(3/2), so 84 is in the sequence.
The terms together with their prime indices begin:
   20: {1,1,3}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   52: {1,1,6}
   56: {1,1,1,4}
   66: {1,2,5}
   68: {1,1,7}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   92: {1,1,9}
   99: {2,2,5}

Crossrefs

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal version is A361856, counted by A361849.
These partitions are counted by A361857, reverse A361858.
Including the equal case gives A361868, counted by A361859.
For mean instead of median we have A361907.
A000975 counts subsets with integer median.
A001222 counts prime factors, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A111907, A237751, A237820, A359893, A361848, A361855, A361908, A361909.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], Max@@prix[#]>2*Median[prix[#]]&]

A361868 Positive integers > 1 whose prime indices satisfy (maximum) >= 2*(median).

Original entry on oeis.org

12, 20, 24, 28, 40, 42, 44, 48, 52, 56, 60, 63, 66, 68, 72, 76, 78, 80, 84, 88, 92, 96, 99, 102, 104, 112, 114, 116, 117, 120, 124, 126, 130, 132, 136, 138, 140, 144, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 189, 190, 192, 195
Offset: 1

Views

Author

Gus Wiseman, Apr 05 2023

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.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 >= 2*(3/2), so 84 is in the sequence.
The terms together with their prime indices begin:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}

Crossrefs

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal case is A361856, counted by A361849.
These partitions are counted by A361859.
The unequal case is A361867, counted by A361857.
The complement is counted by A361858.
A000975 counts subsets with integer median.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
Cf. A027193, A053263, A111907, A118096, A237753, A237821, A361848, A361855, A361906, A361908.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@prix[#]>=2*Median[prix[#]]&]

A361909 Positive integers > 1 whose prime indices satisfy: (maximum) = 2*(length).

Original entry on oeis.org

3, 14, 21, 35, 49, 52, 78, 117, 130, 152, 182, 195, 228, 273, 286, 325, 338, 342, 380, 429, 455, 464, 507, 513, 532, 570, 637, 696, 715, 798, 836, 845, 855, 950, 988, 1001, 1044, 1160, 1183, 1184, 1197, 1254, 1292, 1330, 1425, 1444, 1482, 1566, 1573, 1624
Offset: 1

Views

Author

Gus Wiseman, Apr 05 2023

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.

Examples

			The terms together with their prime indices begin:
     3: {2}
    14: {1,4}
    21: {2,4}
    35: {3,4}
    49: {4,4}
    52: {1,1,6}
    78: {1,2,6}
   117: {2,2,6}
   130: {1,3,6}
   152: {1,1,1,8}
   182: {1,4,6}
   195: {2,3,6}
   228: {1,1,2,8}
   273: {2,4,6}
   286: {1,5,6}
   325: {3,3,6}
   338: {1,6,6}
   342: {1,2,2,8}

Crossrefs

The LHS is A061395 (greatest prime index), least A055396.
Without multiplying by 2 in the RHS, we have A106529.
For omega instead of bigomega we have A111907, counted by A239959.
Partitions of this type are counted by A237753.
The RHS is A255201 (twice bigomega).
For mean instead of length we have A361855, counted by A361853.
For median instead of length we have A361856, counted by A361849.
For minimum instead of length we have A361908, counted by A118096.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors with multiplicity.
A112798 lists prime indices, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A326567/A326568 gives mean of prime indices.
Cf. A067801, A324521, A237752, A237820, A360005, A361205, A361907.

Programs

  • Mathematica
    Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]==2*PrimeOmega[#]&]

A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 0, 5, 0, 6, 3, 5, 0, 11, 6, 8, 7, 10, 0, 36, 0, 14, 16, 16, 29, 43, 0, 21, 36, 69, 0, 97, 0, 35, 138, 33, 0, 150, 61, 137, 134, 74, 0, 231, 134, 265, 229, 56, 0, 650, 0, 65, 749, 267, 247, 533, 0, 405, 565
Offset: 1

Views

Author

Gus Wiseman, Mar 29 2023

Keywords

Comments

Also strict partitions satisfying (maximum) = 2*(mean).
These are strict partitions where both the diagram and its complement (see example) have size n.

Examples

			The a(n) strict partitions for selected n (A..E = 10..14):
  n=9:  n=12:  n=14:  n=15:  n=16:  n=18:  n=20:  n=21:  n=22:
--------------------------------------------------------------
  621   831    7421   A32    8431   C42    A532   E43    B542
        6321          A41    8521   C51    A541   E52    B632
                                    9432   A631   E61    B641
                                    9531   A721          B731
                                    9621   85421         B821
                                           86321
The a(20) = 6 strict partitions are: (10,7,2,1), (10,6,3,1), (10,5,4,1), (10,5,3,2), (8,6,3,2,1), (8,5,4,2,1).
The strict partition y = (8,5,4,2,1) has diagram:
  o o o o o o o o
  o o o o o . . .
  o o o o . . . .
  o o . . . . . .
  o . . . . . . .
Since the partition and its complement (shown in dots) have the same size, y is counted under a(20).

Crossrefs

For minimum instead of mean we have A241035, non-strict A118096.
For length instead of mean we have A241087, non-strict A237753.
For median instead of mean we have A361850, non-strict A361849.
The non-strict version is A361853.
These partitions have ranks A361855 /\ A005117.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A008289 counts strict partitions by length.
A102627 counts strict partitions with integer mean, non-strict A067538.
A116608 counts partitions by number of distinct parts.
A268192 counts partitions by complement size, ranks A326844.
Cf. A111907, A237755, A240850, A326849 A359897, A360068, A360071, A360243, A361848, A361851, A361852, A361906.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[#]*Max@@#==2n&]],{n,30}]
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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