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 15 results. Next

A361849 Number of integer partitions of n such that the maximum is twice the median.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 3, 4, 7, 9, 9, 15, 16, 20, 26, 34, 37, 50, 55, 68, 86, 103, 117, 145, 168, 201, 236, 282, 324, 391, 449, 525, 612, 712, 818, 962, 1106, 1278, 1470, 1698, 1939, 2238, 2550, 2924, 3343, 3824, 4341, 4963, 5627, 6399, 7256, 8231, 9300
Offset: 1

Views

Author

Gus Wiseman, Apr 02 2023

Keywords

Comments

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 a(4) = 1 through a(11) = 9 partitions:
  211  2111  21111  421     422      4221      631        632
                    3211    221111   4311      4222       5321
                    22111   2111111  2211111   42211      5411
                    211111           21111111  322111     42221
                                               2221111    43211
                                               22111111   332111
                                               211111111  22211111
                                                          221111111
                                                          2111111111
For example, the partition (3,2,1,1) has maximum 3 and median 3/2, so is counted under a(7).

Crossrefs

For minimum instead of median we have A118096.
For length instead of median we have A237753.
This is the equal case of A361848.
For mean instead of median we have A361853.
These partitions have ranks A361856.
For "greater" instead of "equal" we have A361857, allowing equality A361859.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
A361860 counts partitions with minimum equal to median.
Cf. A013580, A027193, A053263, A061395, A067659, A111907, A116608, A237755, A240219, A361851.

Programs

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

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)).

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}]

A361855 Numbers > 1 whose prime indices satisfy (maximum) * (length) = 2*(sum).

Original entry on oeis.org

28, 40, 78, 84, 171, 190, 198, 220, 240, 252, 280, 351, 364, 390, 406, 435, 714, 748, 756, 765, 777, 784, 814, 840, 850, 925, 988, 1118, 1197, 1254, 1330, 1352, 1419, 1425, 1440, 1505, 1564, 1600, 1638, 1716, 1755, 1794, 1802, 1820, 1950, 2067, 2204, 2254
Offset: 1

Views

Author

Gus Wiseman, Mar 29 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.
Also positive integers whose prime indices satisfy (maximum) = 2*(mean).
Also Heinz numbers of partitions of the same size as their complement (see example).

Examples

			The terms together with their prime indices begin:
   28: {1,1,4}
   40: {1,1,1,3}
   78: {1,2,6}
   84: {1,1,2,4}
  171: {2,2,8}
  190: {1,3,8}
  198: {1,2,2,5}
  220: {1,1,3,5}
  240: {1,1,1,1,2,3}
  252: {1,1,2,2,4}
  280: {1,1,1,3,4}
The prime indices of 84 are {1,1,2,4}, with maximum 4, length 4, and sum 8, and 4*4 = 2*8, so 84 is in the sequence.
The prime indices of 120 are {1,1,1,2,3}, with maximum 3, length 5, and sum 8, and 3*5 != 2*8, so 120 is not in the sequence.
The prime indices of 252 are {1,1,2,2,4}, with maximum 4, length 5, and sum 10, and 4*5 = 2*10, so 252 is in the sequence.
The partition (5,2,2,1) with Heinz number 198 has diagram:
  o o o o o
  o o . . .
  o o . . .
  o . . . .
Since the partition and its complement (shown in dots) both have size 10, 198 is in the sequence.

Crossrefs

These partitions are counted by A361853, strict A361854.
For median instead of mean we have A361856, counted by A361849.
For minimum instead of mean we have A361908, counted by A118096.
For length instead of mean we have A361909, counted by A237753.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A061395 gives greatest prime index.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
Cf. A067801, A237824, A316413, A324521, A326844, A361205, A361851, A361906.

Programs

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

Formula

A061395(a(n)) * A001222(a(n)) = 2*A056239(a(n)).

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}]

A361907 Number of integer partitions of n such that (length) * (maximum) > 2*n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 4, 7, 11, 19, 26, 43, 60, 80, 115, 171, 201, 297, 374, 485, 656, 853, 1064, 1343, 1758, 2218, 2673, 3477, 4218, 5423, 6523, 7962, 10017, 12104, 14409, 17978, 22031, 26318, 31453, 38176, 45442, 55137, 65775, 77451, 92533, 111485, 131057
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(7) = 3 through a(10) = 11 partitions:
  (511)    (611)     (711)      (721)
  (4111)   (5111)    (5211)     (811)
  (31111)  (41111)   (6111)     (6211)
           (311111)  (42111)    (7111)
                     (51111)    (52111)
                     (411111)   (61111)
                     (3111111)  (421111)
                                (511111)
                                (3211111)
                                (4111111)
                                (31111111)
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 = (4,2,1,1) has length 4 and maximum 4, and 4*4 is not > 2*8, so y is not counted under a(8).
The partition y = (5,1,1,1) has length 4 and maximum 5, and 4*5 > 2*8, so y is counted under a(8).
The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 > 2*9, so y is counted under a(9).
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 A237751, reverse A237754.
For minimum instead of mean we have A237820, reverse A053263.
The complement is counted by A361851, median A361848.
Reversing the inequality gives A361852.
The equal version is A361853.
For median instead of mean we have A361857, reverse A361858.
Allowing equality gives A361906, median A361859.
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.
A116608 counts partitions by number of distinct parts.
A268192 counts partitions by complement size, ranks A326844.
Cf. A027193, A111907, A237752, A237755, A237821, A237824, A237984, A324562, A326622, A327482, A349156, A360071, A361394.

Programs

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

A362046 Number of nonempty subsets of {1..n} with mean n/2.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 9, 8, 25, 23, 75, 68, 235, 213, 759, 695, 2521, 2325, 8555, 7941, 29503, 27561, 103129, 96861, 364547, 344003, 1300819, 1232566, 4679471, 4449849, 16952161, 16171117, 61790441, 59107889, 226451035, 217157068, 833918839, 801467551, 3084255127
Offset: 0

Views

Author

Gus Wiseman, Apr 12 2023

Keywords

Examples

			The a(2) = 1 through a(7) = 8 subsets:
  {1}  {1,2}  {2}      {1,4}      {3}          {1,6}
              {1,3}    {2,3}      {1,5}        {2,5}
              {1,2,3}  {1,2,3,4}  {2,4}        {3,4}
                                  {1,2,6}      {1,2,4,7}
                                  {1,3,5}      {1,2,5,6}
                                  {2,3,4}      {1,3,4,6}
                                  {1,2,3,6}    {2,3,4,5}
                                  {1,2,4,5}    {1,2,3,4,5,6}
                                  {1,2,3,4,5}

Crossrefs

Using range 0..n gives A070925.
Including the empty set gives A133406.
Even bisection is A212352.
For median instead of mean we have A361801, the doubling of A079309.
A version for partitions is A361853, for median A361849.
A000980 counts nonempty subsets of {1..2n-1} with mean n.
A007318 counts subsets by length.
A067538 counts partitions with integer mean, strict A102627.
A231147 appears to count subsets by median, full-steps A013580.
A327475 counts subsets with integer mean, A000975 integer median.
A327481 counts subsets by integer mean.
Cf. A006134, A024718, A057552, A349156, A359893, A361654, A361864.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Mean[#]==n/2&]],{n,0,15}]

Formula

a(n) = (A070925(n) - 1)/2.
a(n) = A133406(n) - 1.
a(2n) = A212352(n) = A000980(n)/2 - 1.

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}]

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[#]&]
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