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.

Previous Showing 21-28 of 28 results.

A111906 Numbers k such that fewer primes, among primes <= the largest prime dividing k, divide k than do not.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 22, 23, 25, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 101, 102, 103, 104, 106, 107, 109, 111, 113, 114, 115
Offset: 1

Views

Author

Leroy Quet, Aug 19 2005

Keywords

Examples

			22 is included because 11 is the largest prime dividing 22. And of the primes <= 11 (2,3,5,7,11), 2 and 11 (2 primes) divide 22, but 3 and 5 and 7 (3 primes) do not divide 22.

Links

Crossrefs

Cf. A111905, A111907.

Programs

  • PARI
    {m=115;v=vector(m);for(n=2,m,f=factor(n)[,1]~;c=0;pc=0;forprime(p=2,vecmax(f), j=1;s=length(f);while(j<=s&&p!=f[j],j++);if(j<=s,c++);pc++);v[n]=sign(pc-2*c)); for(n=1,m,if(v[n]>0,print1(n,",")))} \\ Klaus Brockhaus, Aug 21 2005
  • Python
    from itertools import count, islice
from sympy import sieve, factorint
def a_gen():
    for n in count(3):
        f = [sieve.search(i)[0] for i in factorint(n)]
        if len(f) < (f[-1]+1)//2:
            yield n
A111906_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jun 22 2024

Extensions

More terms from Klaus Brockhaus, Aug 21 2005

A361850 Number of strict integer partitions of n such that the maximum is twice the median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 2, 1, 3, 3, 4, 2, 5, 4, 7, 8, 10, 6, 11, 11, 15, 16, 21, 18, 25, 23, 28, 32, 40, 40, 51, 51, 58, 60, 73, 75, 93, 97, 113, 123, 139, 141, 164, 175, 199, 217, 248, 263, 301, 320, 356, 383, 426, 450, 511, 551, 613, 664, 737
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(7) = 1 through a(20) = 4 strict partitions (A..C = 10..12):
  421  .  .  631  632   .  841   842  843   A51    A52    A53   A54   C62
                  5321     6421       7431  7432   8531   8532  C61   9542
                                      7521  64321  8621         9541  9632
                                                   65321        9631  85421
                                                                9721
The partition (7,4,3,1) has maximum 7 and median 7/2, so is counted under a(15).
The partition (8,6,2,1) has maximum 8 and median 4, so is counted under a(17).

Crossrefs

For minimum instead of median we have A241035, non-strict A237824.
For length instead of median we have A241087, non-strict A237755.
The non-strict version is A361849, ranks A361856.
The non-strict complement is counted by A361857, ranks A361867.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
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.
A359907 counts strict partitions with integer median
A360005 gives median of prime indices (times two), distinct A360457.
Cf. A027193, A067659, A079309, A111907, A116608, A359897, A359908, A360952, A361851, A361858, A361859, A361860.

Programs

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

A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 15, 14, 32, 34, 65, 55, 150, 100, 225, 237, 425, 296, 824, 489, 1267, 1133, 1809, 1254, 4018, 2142, 4499, 4550, 7939, 4564, 14571, 6841, 18285, 16047, 23408, 17495, 52545, 21636, 49943, 51182, 92516, 44582, 144872, 63260, 175318, 169232, 205353
Offset: 0

Views

Author

Gus Wiseman, May 23 2023

Keywords

Comments

Equivalently, n = (length)*(minimum).

Examples

			The a(2) = 1 through a(7) = 14 partitions:
  (31)  (321)  (62)    (32221)  (93)      (3222221)
        (411)  (3221)  (33211)  (552)     (3322211)
               (3311)  (42211)  (642)     (3332111)
               (4211)  (43111)  (732)     (4222211)
               (5111)  (52111)  (822)     (4322111)
                       (61111)  (322221)  (4331111)
                                (332211)  (4421111)
                                (333111)  (5222111)
                                (422211)  (5321111)
                                (432111)  (5411111)
                                (441111)  (6221111)
                                (522111)  (6311111)
                                (531111)  (7211111)
                                (621111)  (8111111)
                                (711111)

Crossrefs

Removing the factor 2 gives A099777.
Taking maximum instead of mean and including odd indices gives A118096.
For length instead of mean and including odd indices we have A237757.
For (maximum) = 2*(mean) see A361851, A361852, A361853, A361854, A361855.
For median instead of mean we have A361861.
These partitions have ranks A363133.
For maximum instead of minimum we have A363218.
For median instead of minimum we have A363224.
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. A053263, A111907, A237753, A237755, A237824, A327482, A349156, A361906, A361907, A363134.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],2*Min@@#==Mean[#]&]],{n,0,15}]
  • Python
    from sympy.utilities.iterables import partitions
def A363132(n): return sum(1 for s,p in partitions(n<<1,m=n,size=True) if n==s*min(p,default=0)) if n else 0 # Chai Wah Wu, Sep 21 2023

Extensions

a(31)-a(46) from Chai Wah Wu, Sep 21 2023

A363134 Positive integers whose multiset of prime indices satisfies: (length) = 2*(minimum).

Original entry on oeis.org

4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 81, 82, 86, 94, 106, 118, 122, 134, 135, 142, 146, 158, 166, 178, 189, 194, 202, 206, 214, 218, 225, 226, 254, 262, 274, 278, 297, 298, 302, 314, 315, 326, 334, 346, 351, 358, 362, 375, 382, 386, 394, 398, 422, 441
Offset: 1

Views

Author

Gus Wiseman, Jun 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:
     4: {1,1}         94: {1,15}       214: {1,28}
     6: {1,2}        106: {1,16}       218: {1,29}
    10: {1,3}        118: {1,17}       225: {2,2,3,3}
    14: {1,4}        122: {1,18}       226: {1,30}
    22: {1,5}        134: {1,19}       254: {1,31}
    26: {1,6}        135: {2,2,2,3}    262: {1,32}
    34: {1,7}        142: {1,20}       274: {1,33}
    38: {1,8}        146: {1,21}       278: {1,34}
    46: {1,9}        158: {1,22}       297: {2,2,2,5}
    58: {1,10}       166: {1,23}       298: {1,35}
    62: {1,11}       178: {1,24}       302: {1,36}
    74: {1,12}       189: {2,2,2,4}    314: {1,37}
    81: {2,2,2,2}    194: {1,25}       315: {2,2,3,4}
    82: {1,13}       202: {1,26}       326: {1,38}
    86: {1,14}       206: {1,27}       334: {1,39}

Crossrefs

Partitions of this type are counted by A237757.
Removing the factor 2 gives A324522.
For maximum instead of length we have A361908, counted by A118096.
For mean instead of length we have A363133, counted by A363132.
For maximum instead of minimum we have A363218, counted by A237753.
A055396 gives minimum prime index, maximum A061395.
A112798 lists prime indices, length A001222, sum A056239.
A360005 gives twice median of prime indices.
Cf. A000961, A006141, A046660, A051293, A106529, A111907, A237755, A237824, A327482, A361860, A361861, A362050.

Programs

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

Formula

A001222(a(n)) = 2*A055396(a(n)).

A362047 Numbers whose prime indices satisfy: (maximum) - (minimum) = (mean).

Original entry on oeis.org

10, 30, 39, 90, 98, 99, 100, 115, 259, 270, 273, 300, 490, 495, 517, 663, 665, 793, 810, 900, 1000, 1083, 1241, 1421, 1495, 1521, 1691, 1911, 2058, 2079, 2125, 2145, 2369, 2430, 2450, 2475, 2662, 2700, 2755, 2821, 3000, 3277, 4247, 4495, 4921, 5587, 5863, 6069
Offset: 1

Views

Author

Gus Wiseman, Apr 11 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:
      10: {1,3}
      30: {1,2,3}
      39: {2,6}
      90: {1,2,2,3}
      98: {1,4,4}
      99: {2,2,5}
     100: {1,1,3,3}
     115: {3,9}
     259: {4,12}
     270: {1,2,2,2,3}
     273: {2,4,6}
     300: {1,1,2,3,3}
The prime indices of 490 are {1,3,4,4}, with minimum 1, maximum 4, and mean 3, and 4-1 = 3, so 490 is in the sequence.

Crossrefs

Partitions of this type are counted by A361862.
For minimum instead of mean we have A361908, counted by A118096.
A055396 gives minimum prime index, A061395 maximum.
A112798 list prime indices, length A001222, sum A056239.
A243055 subtracts the least prime index from the greatest.
A326844 gives the diagram complement size of Heinz partition.
Cf. A111907, A237832, A268192, A316413, A326836, A326837, A326846, A359358, A359360, A361855.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@prix[#]-Min@@prix[#]==Mean[prix[#]]&]
  • Python
    from itertools import count, islice
from sympy import primepi, factorint
def A362047_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(primepi(max(f:=factorint(n)))-primepi(min(f)))*sum(f.values())==sum(primepi(i)*j for i, j in f.items()),count(max(startvalue,2)))
A362047_list = list(islice(A362047_gen(),20)) # Chai Wah Wu, Apr 13 2023

Formula

A359360(a(n)) = A326844(a(n)).
A243055(a(n)) = A061395(a(n)) - A055396(a(n))
= A326567(a(n))/A326568(a(n))
= A056239(a(n))/A001222(a(n)).

A363133 Numbers > 1 whose prime indices satisfy 2*(minimum) = (mean).

Original entry on oeis.org

10, 28, 30, 39, 84, 88, 90, 100, 115, 171, 208, 252, 255, 259, 264, 270, 273, 280, 300, 363, 517, 544, 624, 756, 783, 784, 792, 793, 810, 840, 880, 900, 925, 1000, 1035, 1085, 1197, 1216, 1241, 1425, 1495, 1521, 1595, 1615, 1632, 1683, 1691, 1785, 1872, 1911
Offset: 1

Views

Author

Gus Wiseman, May 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.

Examples

			The terms together with their prime indices begin:
    10: {1,3}
    28: {1,1,4}
    30: {1,2,3}
    39: {2,6}
    84: {1,1,2,4}
    88: {1,1,1,5}
    90: {1,2,2,3}
   100: {1,1,3,3}
   115: {3,9}
   171: {2,2,8}
   208: {1,1,1,1,6}
   252: {1,1,2,2,4}
   255: {2,3,7}
   259: {4,12}
   264: {1,1,1,2,5}

Crossrefs

Removing the factor 2 gives A000961.
For maximum instead of mean we have A361908, counted by A118096.
Partitions of this type are counted by A363132.
For length instead of mean we have A363134, counted by A237757.
For 2*(maximum) = (length) we have A363218, counted by A237753.
A051293 counts subsets with integer mean.
A112798 lists prime indices, length A001222, sum A056239.
A360005 gives twice median of prime indices.
Cf. A006141, A106529, A111907, A237755, A237824, A324522, A327482, A361860, A361861, A362050.

Programs

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

A363222 Numbers whose multiset of prime indices satisfies (maximum) - (minimum) = (length).

Original entry on oeis.org

10, 21, 28, 42, 55, 70, 88, 91, 98, 99, 132, 165, 187, 198, 208, 220, 231, 247, 308, 312, 325, 330, 351, 363, 391, 455, 462, 468, 484, 520, 544, 550, 551, 585, 702, 713, 715, 726, 728, 770, 780, 816, 819, 833, 845, 975, 1073, 1078, 1092, 1144, 1170, 1210, 1216
Offset: 1

Views

Author

Gus Wiseman, May 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.

Examples

			The terms together with their prime indices begin:
    10: {1,3}
    21: {2,4}
    28: {1,1,4}
    42: {1,2,4}
    55: {3,5}
    70: {1,3,4}
    88: {1,1,1,5}
    91: {4,6}
    98: {1,4,4}
    99: {2,2,5}
   132: {1,1,2,5}
   165: {2,3,5}
   187: {5,7}
   198: {1,2,2,5}

Crossrefs

The RHS is A001222.
Partitions of this type are counted by A237832.
The LHS (maximum minus minimum) is A243055.
A001221 (omega) counts distinct prime factors.
A112798 lists prime indices, sum A056239.
A360005 gives median of prime indices, distinct A360457.
Cf. A067801, A111907, A118096, A237821, A361205, A361908, A361909.

Programs

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

Formula

A061395(a(n)) - A055396(a(n)) = A001222(a(n)).

A363221 Number of strict integer partitions of n such that (length) * (maximum) <= 2n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 19, 23, 26, 29, 37, 39, 49, 55, 62, 71, 84, 93, 108, 118, 141, 149, 188, 193, 217, 257, 279, 318, 369, 376, 441, 495, 572, 587, 692, 760, 811, 960, 1046, 1065, 1307, 1387, 1550, 1703, 1796, 2041, 2295, 2456, 2753, 3014
Offset: 1

Views

Author

Gus Wiseman, May 23 2023

Keywords

Comments

Also strict partitions such that (maximum) <= 2*(mean).
These are strict partitions whose complement (see A361851) has size <= n.

Examples

			The partition y = (4,3,1) has length 3 and maximum 4, and 3*4 <= 2*8, so y is counted under a(8). The complement of y has size 4, which is less than or equal to n = 8.

Crossrefs

The equal case for median is A361850, non-strict A361849 (ranks A361856).
The non-strict version is A361851, A361848 for median.
The equal case is A361854, non-strict A361853 (ranks A361855).
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, A241061, A241086, A324521, A324562, A349156, A360068, A360241, A361394, A361852, A361906.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Max@@#<=2*Mean[#]&]],{n,30}]
Previous Showing 21-28 of 28 results.

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