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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.

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A343338 Numbers with no prime index dividing or divisible by all the other prime indices.

Original entry on oeis.org

1, 15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 91, 93, 95, 99, 105, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 201, 203, 205, 207, 209, 215, 217, 219, 221, 225, 231, 245, 247, 249, 253, 255, 265, 275, 279, 285, 287, 291, 295, 297, 299, 301
Offset: 1

Views

Author

Gus Wiseman, Apr 13 2021

Keywords

Comments

Alternative name: 1 and numbers whose smallest prime index does not divide all the other prime indices, nor whose greatest prime index is divisible by all the other prime indices.
First differs from A302697 in having 91.
First differs from A337987 in having 91.
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 Heinz numbers of partitions with greatest part not divisible by all the others and smallest part not dividing all the others (counted by A343342). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}         105: {2,3,4}      203: {4,10}
     15: {2,3}      119: {4,7}        205: {3,13}
     33: {2,5}      123: {2,13}       207: {2,2,9}
     35: {3,4}      135: {2,2,2,3}    209: {5,8}
     45: {2,2,3}    141: {2,15}       215: {3,14}
     51: {2,7}      143: {5,6}        217: {4,11}
     55: {3,5}      145: {3,10}       219: {2,21}
     69: {2,9}      153: {2,2,7}      221: {6,7}
     75: {2,3,3}    155: {3,11}       225: {2,2,3,3}
     77: {4,5}      161: {4,9}        231: {2,4,5}
     85: {3,7}      165: {2,3,5}      245: {3,4,4}
     91: {4,6}      175: {3,3,4}      247: {6,8}
     93: {2,11}     177: {2,17}       249: {2,23}
     95: {3,8}      187: {5,7}        253: {5,9}
     99: {2,2,5}    201: {2,19}       255: {2,3,7}
For example, the prime indices of 975 are {2,3,3,6}, all of which divide 6, but not all of which are multiples of 2, so 975 is not in the sequence.

Crossrefs

The first condition alone gives A342193.
The second condition alone gives A343337.
The half-opposite versions are A343339 and A343340.
The partitions with these Heinz numbers are counted by A343342.
The opposite version is the complement of A343343.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
A056239 adds up prime indices, row sums of A112798.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
A339564 counts factorizations with a selected factor.
Cf. A083710, A130689, A338470, A339562, A341450, A343341, A343346, A343347, A343348, A343377, A343379, A343382.

Programs

  • Mathematica
    Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)&&!And@@IntegerQ/@(p/Min@@p)]&]

Formula

Intersection of A342193 and A343337.

A343380 Number of strict integer partitions of n with no part dividing all the others but with a part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 1, 4, 0, 1, 0, 2, 0, 4, 0, 3, 1, 2, 2, 5, 0, 5, 3, 4, 1, 9, 1, 5, 2, 4, 5, 11, 1, 6, 4, 11, 3, 13, 5, 10, 4, 11, 8, 14, 3, 10, 6, 9, 3, 15, 6, 14, 10, 18, 8
Offset: 0

Views

Author

Gus Wiseman, Apr 16 2021

Keywords

Comments

Alternative name: Number of strict integer partitions of n that are either empty or (1) have smallest part not dividing all the others and (2) have greatest part divisible by all the others.

Examples

			The a(11) = 1 through a(29) = 4 partitions (empty columns indicated by dots, A..O = 10..24):
  632  .  .  .  .  .  A52  .  C43  .  C432  C64  E72   .  C643  .  K52    .  I92
                      C32                        F53               C6432     K54
                                                 I32                         O32
                                                 C632                        I632

Crossrefs

The first condition alone gives A341450.
The non-strict version is A343344 (Heinz numbers: A343339).
The second condition alone gives A343347.
The half-opposite versions are A343378 and A343379.
The opposite (and dual) version is A343381.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A097986, A130689, A200745, A264401, A338470, A339562, A342193, A343377, A343382.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

A343344 Number of integer partitions of n that are either empty, or do not have smallest part dividing all the others, but do have greatest part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 5, 1, 6, 4, 6, 7, 15, 6, 16, 15, 20, 17, 36, 18, 43, 36, 46, 48, 72, 45, 93, 82, 103, 88, 152, 104, 179, 158, 191, 194, 285, 202, 328, 292, 373, 348, 502, 391, 576, 519, 659, 634, 864, 665
Offset: 0

Views

Author

Gus Wiseman, Apr 15 2021

Keywords

Comments

Alternative name: Number of integer partitions of n with no part dividing all the others, but with a part divisible by all the others.

Examples

			The a(18) = 1  through a(23) = 15 partitions (A..E = 10..14):
  633222   C43       C332      C432       C64        E72
           A522      66332     A5222      A552       F53
           C322      633332    C3222      C433       I32
           66322     6332222   663222     C3322      C443
           633322              6333222    663322     C632
           6322222             63222222   6333322    66632
                                          63322222   C3332
                                                     C4322
                                                     663332
                                                     A52222
                                                     C32222
                                                     6333332
                                                     6632222
                                                     63332222
                                                     632222222

Crossrefs

The second condition alone gives A130689.
The half-opposite versions are A130714 and A343342.
The first condition alone gives A338470.
The Heinz numbers of these partitions are 1 and A343339.
The opposite version is A343345.
The strict case is A343380.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
Cf. A083710, A097986, A264401, A339562, A341450, A342193, A343346.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.

Original entry on oeis.org

1, 15, 30, 33, 35, 45, 51, 55, 60, 66, 69, 70, 75, 77, 85, 90, 91, 93, 95, 99, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 150, 153, 154, 155, 161, 165, 170, 175, 177, 180, 182, 186, 187, 190, 195, 198, 201, 203, 204, 205, 207, 209, 210
Offset: 1

Views

Author

Gus Wiseman, Apr 15 2021

Keywords

Comments

After 1, first differs from A318992 in lacking 390, with prime indices {1,2,3,6}.
First differs from A343337 in having 195, with prime indices {2,3,6}.
Alternative name: 1 and numbers where either the smallest prime index is not a divisor of all the other prime indices, or the greatest prime index is not divisible by all the other prime indices.
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 Heinz numbers of partitions that either empty, have smallest part not dividing all the others, or have greatest part not divisible by all the others (counted by A343346). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}            90: {1,2,2,3}      141: {2,15}
     15: {2,3}         91: {4,6}          143: {5,6}
     30: {1,2,3}       93: {2,11}         145: {3,10}
     33: {2,5}         95: {3,8}          150: {1,2,3,3}
     35: {3,4}         99: {2,2,5}        153: {2,2,7}
     45: {2,2,3}      102: {1,2,7}        154: {1,4,5}
     51: {2,7}        105: {2,3,4}        155: {3,11}
     55: {3,5}        110: {1,3,5}        161: {4,9}
     60: {1,1,2,3}    119: {4,7}          165: {2,3,5}
     66: {1,2,5}      120: {1,1,1,2,3}    170: {1,3,7}
     69: {2,9}        123: {2,13}         175: {3,3,4}
     70: {1,3,4}      132: {1,1,2,5}      177: {2,17}
     75: {2,3,3}      135: {2,2,2,3}      180: {1,1,2,2,3}
     77: {4,5}        138: {1,2,9}        182: {1,4,6}
     85: {3,7}        140: {1,1,3,4}      186: {1,2,11}
For example, the prime indices of 90 are {1,2,2,3}, and, because 1 divides all the other parts, 90 is in the sequence, even though 3 is not divisible by all the other parts.

Crossrefs

The partitions without these Heinz numbers are counted by A130714.
The first condition alone gives A342193.
The second condition alone gives A343337.
The "and" instead of "or" version is A343338.
The partitions with these Heinz numbers are counted by A343346.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A056239 adds up prime indices, row sums of A112798.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
A339564 counts factorizations with a selected factor.
Cf. A083710, A257993, A338470, A339562, A341450, A343339, A343340, A343341, A343342, A343378, A343379, A343382.

Programs

  • Mathematica
    Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)||!And@@IntegerQ/@(p/Min@@p)]&]

Formula

Equals the union of A342193 and A343337.

A343340 Numbers with a prime index dividing all the other prime indices, but with no prime index divisible by all the other prime indices.

Original entry on oeis.org

30, 60, 66, 70, 90, 102, 110, 120, 132, 138, 140, 150, 154, 170, 180, 182, 186, 190, 198, 204, 210, 220, 238, 240, 246, 264, 270, 273, 276, 280, 282, 286, 290, 300, 306, 308, 310, 322, 330, 340, 350, 354, 360, 364, 372, 374, 380, 396, 402, 406, 408, 410, 414
Offset: 1

Views

Author

Gus Wiseman, Apr 15 2021

Keywords

Comments

Alternative name: Numbers > 1 whose smallest prime index divides all the other prime indices, but whose greatest prime index is not divisible by all the other prime indices.
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 Heinz numbers of partitions with greatest part not divisible by all the others, but smallest part dividing all the others (counted by A343345). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of terms together with their prime indices begins:
     30: {1,2,3}        182: {1,4,6}          282: {1,2,15}
     60: {1,1,2,3}      186: {1,2,11}         286: {1,5,6}
     66: {1,2,5}        190: {1,3,8}          290: {1,3,10}
     70: {1,3,4}        198: {1,2,2,5}        300: {1,1,2,3,3}
     90: {1,2,2,3}      204: {1,1,2,7}        306: {1,2,2,7}
    102: {1,2,7}        210: {1,2,3,4}        308: {1,1,4,5}
    110: {1,3,5}        220: {1,1,3,5}        310: {1,3,11}
    120: {1,1,1,2,3}    238: {1,4,7}          322: {1,4,9}
    132: {1,1,2,5}      240: {1,1,1,1,2,3}    330: {1,2,3,5}
    138: {1,2,9}        246: {1,2,13}         340: {1,1,3,7}
    140: {1,1,3,4}      264: {1,1,1,2,5}      350: {1,3,3,4}
    150: {1,2,3,3}      270: {1,2,2,2,3}      354: {1,2,17}
    154: {1,4,5}        273: {2,4,6}          360: {1,1,1,2,2,3}
    170: {1,3,7}        276: {1,1,2,9}        364: {1,1,4,6}
    180: {1,1,2,2,3}    280: {1,1,1,3,4}      372: {1,1,2,11}

Crossrefs

The first condition alone gives the complement of A342193.
The second condition alone gives A343337.
The partitions with these Heinz numbers are counted by A343345.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
A056239 adds up prime indices, row sums of A112798.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
Cf. A083710, A083711, A097986, A098965, A130714, A339562, A339563, A343341, A343377, A343381.

Programs

  • Mathematica
    Select[Range[2,100],With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)&&And@@IntegerQ/@(p/Min@@p)]&]

Formula

Complement of A342193 in A343337.

A343339 Numbers with no prime index dividing all the other prime indices, but with a prime index divisible by all the other prime indices.

Original entry on oeis.org

195, 555, 585, 915, 957, 975, 1295, 1335, 1665, 1695, 1755, 2193, 2265, 2343, 2535, 2585, 2715, 2745, 2775, 2871, 2925, 3115, 3345, 3367, 3729, 3765, 3885, 4005, 4209, 4215, 4575, 4755, 4875, 4995, 5085, 5265, 5285, 5385, 5457, 5467, 5709, 5955, 6205, 6215
Offset: 1

Views

Author

Gus Wiseman, Apr 15 2021

Keywords

Comments

Alternative name: Numbers > 1 whose smallest prime index does not divide all the other prime indices, but whose greatest prime index is divisible by all the other prime indices.
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 Heinz numbers of partitions with greatest part divisible by all the others, but smallest part not dividing all the others (counted by A343344). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of terms together with their prime indices begins:
     195: {2,3,6}        2585: {3,5,15}       4575: {2,3,3,18}
     555: {2,3,12}       2715: {2,3,42}       4755: {2,3,66}
     585: {2,2,3,6}      2745: {2,2,3,18}     4875: {2,3,3,3,6}
     915: {2,3,18}       2775: {2,3,3,12}     4995: {2,2,2,3,12}
     957: {2,5,10}       2871: {2,2,5,10}     5085: {2,2,3,30}
     975: {2,3,3,6}      2925: {2,2,3,3,6}    5265: {2,2,2,2,3,6}
    1295: {3,4,12}       3115: {3,4,24}       5285: {3,4,36}
    1335: {2,3,24}       3345: {2,3,48}       5385: {2,3,72}
    1665: {2,2,3,12}     3367: {4,6,12}       5457: {2,7,28}
    1695: {2,3,30}       3729: {2,5,30}       5467: {4,5,20}
    1755: {2,2,2,3,6}    3765: {2,3,54}       5709: {2,5,40}
    2193: {2,7,14}       3885: {2,3,4,12}     5955: {2,3,78}
    2265: {2,3,36}       4005: {2,2,3,24}     6205: {3,7,21}
    2343: {2,5,20}       4209: {2,9,18}       6215: {3,5,30}
    2535: {2,3,6,6}      4215: {2,3,60}       6475: {3,3,4,12}

Crossrefs

The first condition alone gives A342193.
The second condition alone gives the complement of A343337.
The partitions with these Heinz numbers are counted by A343344.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
A339564 counts factorizations with a selected factor.
Cf. A130689, A130714, A257993, A338470, A339562, A341450, A343338, A343380.

Programs

  • Mathematica
    Select[Range[2,1000],With[{p=PrimePi/@First/@FactorInteger[#]},And@@IntegerQ/@(Max@@p/p)&&!And@@IntegerQ/@(p/Min@@p)]&]

Formula

Complement of A343337 in A342193.
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